Comma Sums
(plus
a tree
by Aai)
J’ai envoyé ceci à la liste SeqFan le 4 octobre 2011 sous le titre (ampoulé) « First
differences are the sums of the two digits touching the commas » :
> Hello SeqFans,
>
> S =
0,5,11,13,17,26,35,44,53,62,71,79,97,105,111,113,117,...
>
> Consider the first comma; its closest digits are 0 and 5;
> add 0 to 5; the sum (5) is the first difference between 0 and 5.
>
> Consider the second comma; its closest digits are 5 and 1;
> add 5 to 1; the sum (6) is the first difference between 5 and 11.
> Etc.
>
> Does S
stop at some point? If yes, could another -longer- S arise from a different start?
>
> Best,
> É.
Jack Brennen répondit quelques laps plus tard :
This sequence stops after 123 terms. The 123rd term is 989
which has no possible successor.
It looks like many low starting points lead to 989 and halt there.
The smallest starting point which goes
beyond that seems to be the number 396 (which has no
possible predecessor).
The sequence beginning with 396 goes on for quite
some distance.
I traced it up to values in excess of 20 million (over 3 million terms).
[Éric]
to Jack Brennen:
Thank you, Jack -- this is exactly
what I was looking for,
Best,
É.
(some
integers can be followed by two integers, as you will have noticed:
14 --> 19 or 20
28 --> 39 or 40
... but this doesn’t change much)
Lars Blomberg:
Hello Eric,
As for 396, I notice that the
first differences of the sequence are:
396,10,10,10,10,10,10,10,10,10,10,11,12,14,8,6,12,14,8,6,12,15,
10,10,10,10,10,10,10,10,11,12,14,8,16,12,14,8,17,14,8,16,12,14,
8,16,13,16,12,14,18,16,12,(6,2,4,8)
... repeated
indefinitely, which suggests that the sequence is infinite.
This looks like an interesting class of
sequences (with a simple formulation!)
More investigations will follow, if you are
interested.
Regards,
Lars
-----------
(...)
Hello again,
I immediately discovered that the first
differences starting from 0,5 are
5,6,2,4,9,9,9,9,9,9,8,18,8,
6,2,4,8,
6,2,4,8,
6,2,4,8,
6,2,4,8,
6,2,4,9,8,
6,2,4,8,
6,2,4,8,
6,2,4,8,
6,2,4,8,
7,4,8,
6,12,4,8,
6,12,4,8,
6,12,4,9,8,
6,12,4,8,
6,12,4,8,
6,12,4,9,8,
6,12,14,8,6,12,14,8,6,13,6,12,14,8,6,12,14,8,6,13,16,12,14,8,16,12,15,
10,10,10,10,10,10,10,10,10,11,12,14,18,16,12,14
So repetitions of 6,2,4,8 may occur and still
give a finite sequence.
More to investigate...
----------
(...)
Hello Eric,
Iterating 396 for a while I
reached the number 101,709,259,615 after 12,490,000,000 iterations.
Further:
2.1
Numbers with no successors
Empirically for number of digits n=2,3,4:
86
986 9986
87
987 9987
88
988 9988
89
989 9989
91 991
9991
92
992 9992
93
993 9993
94
994 9994
It seems that the rule is that numbers of the
form:
10^n - s, s=14, 13, 12, 11, 9, 8, 7, 6
have no successors. Can this be proven? It suggests
that once an iteration has come into the millions, the
"chance" of hitting a no-successor becomes very small.
2.2
Numbers with 2 successors
The first few numbers with 2
successors.
9
19 20
33
39 40
47
59 60
71
79 80
189
199 200
393
399 400
587
599 600
791 799 800
1989
1999 2000
3993
3999 4000
5987
5999 6000
7991
7999 8000
19989
19999 20000
39993
39999 40000
59987
59999 60000
79991
79999 80000
199989
199999 200000
399993
399999 400000
599987
599999 600000
799991
799999 800000
1999989
1999999 2000000
3999993
3999999 4000000
5999987
5999999 6000000
7999991
7999999 8000000
19999989
19999999 20000000
39999993
39999999 40000000
59999987
59999999 60000000
79999991
79999999 80000000
The rule seems to be that one of the successors
start with an odd digit followed by all 9’s, the other starts with the next
even digit followed by all 0’s. Can this be proven? From this it follows that
numbers with 2 successors are of the form:
Predecessor
Successor 1 Successor 2
2*10^n-11
1*10^n-1 2*10^n
4*10^n-7
3*10^n-1 4*10^n
6*10^n-13
5*10^n-1 6*10^n
8*10^n-9
7*10^n-1 8*10^n
(No number seems to have more than 3
successors. Can this be proven?)
Continuing...
/Lars
Aai (private
mail):
Hi Eric,
here are some more numbers with 2 successors. The
pattern is obvious I think (if not already known by you) :-)
9 14 28
33 47 52
66 71
189 194
388 393 587
592 786 791
1989 1994 3988
3993 5987 5992
7986 7991
19989
19994 39988 39993 59987 59992 79986 79991
Mark Brader (on rec.puzzles):
This computes the two versions of the series starting
with 0, 5:
use
warnings;
use
strict;
sub
succ {
my ($x) = @_;
my $end = $x % 10;
my @ok;
return map {
my $nextx = $x + $end
+ $_;
($nextx =~ /^$_/) ?
($nextx) : ()
} (1 .. 9);
}
my
$i = 0;
print "[", ++$i,
"] 0\n";
for
(my @curr = (5); @curr; @curr = map { succ($_) } @curr) {
print
"[", ++$i, "] @curr\n";
}
The results are:
[1] 0
[2] 5
[3] 11
[4] 13
[5] 17
[6] 26
[7] 35
[8] 44
[9] 53
[10] 62
[11] 71
[12] 79 80
[13] 97 88
[14] 105
[15] 111
[16] 113
[17] 117
[18] 125
[19] 131
[20] 133
[21] 137
[22] 145
[23] 151
[24] 153
[25] 157
[26] 165
[27] 171
[28] 173
[29] 177
[30] 185
[31] 191
[32] 193
[33] 197
[34] 206
[35] 214
[36] 220
[37] 222
[38] 226
[39] 234
[40] 240
[41] 242
[42] 246
[43] 254
[44] 260
[45] 262
[46] 266
[47] 274
[48] 280
[49] 282
[50] 286
[51] 294
[52] 301
[53] 305
[54] 313
[55] 319
[56] 331
[57] 335
[58] 343
[59] 349
[60] 361
[61] 365
[62] 373
[63] 379
[64] 391
[65] 395
[66] 404
[67] 412
[68] 418
[69] 430
[70] 434
[71] 442
[72] 448
[73] 460
[74] 464
[75] 472
[76] 478
[77] 490
[78] 494
[79] 503
[80] 511
[81] 517
[82] 529
[83] 543
[84] 551
[85] 557
[86] 569
[87] 583
[88] 591
[89] 597
[90] 610
[91] 616
[92] 628
[93] 642
[94] 650
[95] 656
[96] 668
[97] 682
[98] 690
[99] 696
[100] 709
[101] 725
[102] 737
[103] 751
[104] 759
[105] 775
[106] 787
[107] 802
[108] 812
[109] 822
[110] 832
[111] 842
[112] 852
[113] 862
[114] 872
[115] 882
[116] 892
[117] 903
[118] 915
[119] 929
[120] 947
[121] 963
[122] 975
[123] 989
To compute all possible versions of the series
starting from 0, change the last part to:
my
$i = 0;
for
(my @curr = (0); @curr; @curr = map { succ($_) } @curr) {
print
"[", ++$i, "] @curr\n";
}
And you still get 123 lines, ending with:
[114] 872 882 882 892 892 882
[115] 882 892 892 903 903 892
[116] 892 903 903 915 915 903
[117] 903 915 915 929 929 915
[118] 915 929 929 947 947 929
[119] 929 947 947 963 963 947
[120] 947 963 963 975 975 963
[121] 963 975 975 989 989 975
[122] 975 989 989 989
[123] 989
The program was written for ease of coding, not for
speed -- I wouldn’t’ve used Perl if I was going for
speed. But, changing (0) to (396), I find that in
2 minutes it computes about 1/4 million terms. I’ll start it running now and
come back later and see what happens.
(...)
I killed it when it had used 1 hour of CPU time. The
sequence began:
[1] 396
[2] 406
[3] 416
[4] 426
[5] 436
[6] 446
[7] 456
[8] 466
[9] 476
[10] 486
[11] 496
[12] 507
[13] 519
[14] 533
[15] 541
[16] 547
[17] 559
[18] 573
[19] 581
[20] 587
[21] 599 600
The two versions eventually came back together with a
1-step offset:
[250] 1981 1973
[251] 1983 1977
[252] 1987 1985
[253] 1995 1991
[254] 2002 1993
[255] 2006 1997
[256] 2014 2006
That is, 2,006 was the 255th or 256th term depending
on which choice was taken for the 21st term.
Because I hadn’t programmed to check for that
situation, it went on computing both versions of the series until I killed it.
There were no further bifurcations and the last value computed and written out
was 52,531,251, which was the 8,134,537th or 8,134,538th term. I was curious
enough about this to write a C version of the program, designed to terminate if
the series bifurcated again, ended, or exceeded a value of 1 billion. I
checked it on the starting values 5 and 396 and got the
results already seen.
Then I started it on a value of 2006:
[1] 2006
[2] 2014
[3] 2020
[4] 2022
[5] 2026
[6] 2034
[7] 2040
[8] 2042
[9] 2046
[10] 2054
[11] 2060
[12] 2062
[13] 2066
[14] 2074
[15] 2080
[16] 2082
[17] 2086
[18] 2094
[19] 2100
[20] 2102
[21] 2106
[22] 2114
[23] 2120
and several hours and about 2
gigabytes of disk space later, it terminated due to passing 1 billion:
[121481135] 999999667
[121481136] 999999683
[121481137] 999999695
[121481138] 999999709
[121481139] 999999727
[121481140] 999999743
[121481141] 999999755
[121481142] 999999769
[121481143] 999999787
[121481144] 999999803
[121481145] 999999815
[121481146] 999999829
[121481147] 999999847
[121481148] 999999863
[121481149] 999999875
[121481150] 999999889
[121481151] 999999907
[121481152] 999999923
[121481153] 999999935
[121481154] 999999949
[121481155] 999999967
[121481156] 999999983
[121481157] 999999995
So, starting from 396 as the first
term, we will reach 1,000,000,001 as the 121,481,412th or 121,481,413th term,
and the series is still going, with no more bifurcations after the one at the
21st term.
Here’s the program:
#include <stdio.h>
#include <stdlib.h>
int main (int
c, char **v)
{
long
i, n, succ (long);
if
(c != 2) {
fprintf (stderr, "Usage\n");
return 1;
}
n = atol
(v[1]);
for
(i = 1; n >= 0; ++i, n = succ (n))
printf ("[%d]
%d\n", i, n);
return
0;
}
long succ
(long n)
{
long
n_plus_end = n + n % 10;
int dig, start;
long
next1 = -1, next2;
int dig1 (long);
for
(dig = 1; dig < 10; ++dig) {
next2 = n_plus_end + dig;
start = dig1 (next2);
if (dig == start) {
if (next1 > 0) {
printf ("Bifurcation: %ld
-> %ld or %ld\n",
n, next1, next2);
return -1;
}
next1 = next2;
}
}
if
(next1 < 0) printf ("End\n");
return
next1;
}
int dig1 (long n)
{
/* I think this should be
faster than doing / in a loop */
if
(n < 10000) {
if (n < 100) {
if (n < 10)
return n;
else return n/10;
} else {
if (n < 1000)
return n/100;
else return n/1000;
}
} else if (n < 100000000)
{
if (n < 1000000) {
if (n < 100000)
return n/10000;
else return
n/100000;
} else {
if (n <
10000000) return n/1000000;
else return
n/10000000;
}
} else if (n <
1000000000) return n/100000000;
else
{
printf ("Cannot
handle n = %ld\n", n);
exit (1);
}
}
Nicolas Graner:
I am surprised that you haven’t noticed that
the sequence starting with 396 is infinite.
Starting from 1001 (which is reached from 396), you eventually reach 9995. The next number is 10001,
which eventually yields 99995, followed by 100001. Notice a pattern? :)
I haven’t written a formal proof, but it should
be pretty easy, by induction over the number of digits. Here is the idea:
Take 1x001, where x is any sequence of digits.
It will reproduce the sequence from 1001 to 1995, up to 1x995. This is followed
by 1y001, with y = X+1. This repeats until x becomes all 9’s, at which point
the first digit will change: 199..995 -> 200..002.
Now we get numbers of the form 2x002, which
reproduce the sequence from 2002 to 2994 while x increases, until it reaches
all 9’s and the first digit becomes 3.
You can repeat this reasoning with each value
of the first digit. There are some subtleties because, when the first digit
changes, the last one is not always the same. But since the sequence is very
"robust" (as evidenced by forking sequences, which reunite pretty
quickly), variations in the last digit have no effect beyond a couple of terms.
So, starting with 100*001, we are bound to get
eventually to 999*995, which is followed with 1000*001 (with one more digit). Which shows that the sequence is infinite.
OK, mathematically speaking I
can’t claim the sequence is infinite until someone works out every detail of
the proof, but I’d be really surprised it the "obvious" pattern didn’t
repeat indefinitely.
Nicolas
Jack Brennen:
I figured that it probably was infinite, but
proving it requires a whole lot more time than stating it. :)
An outline of a proof would be to prove that
each of the following nine relationships is valid, where z represents some
number of consecutive zeroes (possibly including no zeroes -- the empty
string):
10z01 leads to 20z02
20z02 leads to 30z01
30z01 leads to 40z04
40z04 leads to 50z03
50z03 leads to 60z04
60z04 leads to 70z05
70z05 leads to 80z06
80z06 leads to 90z07
90z07 leads to 100z01
Jean-Marc Falcoz:
Salut Eric,
Inutile de te dire que
j’aime bien cette suite ! Je n’ai pas tenu compte
des cas où
un nombre a deux successeurs possibles, j’ai pris le plus petit qui marche. Voici donc
deux ou trois
résultats pêle-mêle :
1) le graphe de la suite commençant par
396 (un peu plus de
200000 termes).
Comme souvent avec les
suites "presque" périodiques,
un graphe illustre bien la situation : on voit ces espèces
de chaînettes, vers 12000,
120000, et 1200000 (je suppose)
2) Une liste de nombre sans successeurs :
{1, 2, 3, 4, 86, 87, 88, 89, 91, 92, 93, 94, 986, 987, 988, 989, 991,
992, 993, 994, 9987, 9989, 9991, 9993}
3) une liste partielle (1 à 2000, puis 8000 à
9000) avec le premier terme, le dernier (si la suite s’arrête, sinon le premier terme dépassant 1000000), et le nombre
de termes (surtout utile si la suite s’arrête)
{{1, 1, 1}, {2, 2, 1}, {3, 3, 1}, {4, 4, 1},
{5, 989, 122}, {6, 989, 121}, {7, 989, 121}, {8, 989, 120}, {9, 989, 120}, {10,
989, 122}, {11, 989, 121}, {12, 989, 121}, {13, 989, 120}, {14, 989, 120}, {15,
989, 120}, {16, 989, 120}, {17, 989, 119}, {18, 989, 118}, {19, 989, 119}, {20,
989, 120}, {21, 989, 120}, {22, 989, 119}, {23, 989, 118}, {24, 989, 119}, {25,
989, 118}, {26, 989, 118}, {27, 989, 117}, {28, 989, 117}, {29, 989, 117}, {30,
989, 118}, {31, 989, 118}, {32, 989, 117}, {33, 989, 117}, {34, 989, 117}, {35,
989, 117}, {36, 989, 116}, {37, 989, 116}, {38, 989, 116}, {39, 989, 116}, {40,
989, 117}, {41, 989, 116}, {42, 989, 116}, {43, 989, 116}, {44, 989, 116}, {45,
989, 115}, {46, 989, 115}, {47, 88, 4}, {48, 989, 115}, {49, 88, 4}, {50, 989,
115}, {51, 989, 115}, {52, 88, 4}, {53,
989, 115}, {54, 88, 4}, {55, 989, 114}, {56, 989, 114}, {57, 989, 114}, {58,
989, 114}, {59, 88, 3}, {60, 989, 114}, {61, 989, 114}, {62, 989, 114}, {63,
989, 114}, {64, 88, 3}, {65, 93, 3},
{66, 989, 113}, {67, 93, 3}, {68, 989, 113}, {69, 86, 2}, {70, 93, 3}, {71,
989, 113}, {72, 93, 3}, {73, 989, 113}, {74, 86, 2}, {75, 88, 2}, {76, 91, 2},
{77, 93, 2}, {78, 989, 113}, {79, 989, 112}, {80, 88, 2}, {81, 91, 2}, {82, 93,
2}, {83, 989, 113}, {84, 989, 112}, {85, 989, 102}, {86, 86, 1}, {87, 87, 1},
{88, 88, 1}, {89, 89, 1}, {90, 989, 102}, {91, 91, 1}, {92, 92, 1}, {93, 93,
1}, {94, 94, 1}, {95, 989, 112}, {96, 989,
111}, {97, 989, 111}, {98, 989, 110}, {99, 989, 101}, {100, 989, 112},
{101, 989, 111}, {102, 989, 111}, {103, 989, 110}, {104, 989, 101}, {105, 989, 110},
{106, 989, 109}, {107, 989, 109}, {108, 989, 108}, {109, 989, 100}, {110, 989,
110}, {111, 989, 109}, {112, 989, 109}, {113, 989, 108}, {114, 989, 100}, {115,
989, 108}, {116, 989, 107}, {117, 989, 107}, {118, 989, 106}, {119, 989, 99},
{120, 989, 108}, {121, 989, 107}, {122, 989, 107}, {123, 989, 106}, {124, 989,
99}, {125, 989, 106}, {126, 989, 105}, {127, 989, 105}, {128, 989, 104}, {129,
989, 98}, {130, 989, 106}, {131, 989, 105}, {132, 989, 105}, {133, 989, 104},
{134, 989, 98}, {135, 989, 104}, {136, 989, 103}, {137, 989, 103}, {138, 989,
102}, {139, 989, 97}, {140, 989, 104}, {141, 989, 103}, {142, 989, 103}, {143,
989, 102}, {144, 989, 97}, {145, 989, 102}, {146, 989, 101}, {147, 989, 101},
{148, 989, 100}, {149, 989, 96}, {150, 989, 102}, {151, 989, 101}, {152, 989,
101}, {153, 989, 100}, {154, 989, 96}, {155, 989, 100}, {156, 989, 99}, {157,
989, 99}, {158, 989, 98}, {159, 989, 95}, {160, 989, 100}, {161, 989, 99},
{162, 989, 99}, {163, 989, 98}, {164, 989, 95}, {165, 989, 98}, {166, 989, 97},
{167, 989, 97}, {168, 989, 96}, {169, 989, 94}, {170, 989, 98}, {171, 989, 97},
{172, 989, 97}, {173, 989, 96}, {174, 989, 94}, {175, 989, 96}, {176, 989, 95},
{177, 989, 95}, {178, 989, 94}, {179, 989, 93}, {180, 989, 96}, {181, 989, 95},
{182, 989, 95}, {183, 989, 94}, {184, 989, 93}, {185, 989, 94}, {186, 989, 93},
{187, 989, 93}, {188, 989, 92}, {189, 989, 92}, {190, 989, 94}, {191, 989, 93},
{192, 989, 93}, {193, 989, 92}, {194, 989, 92}, {195, 989, 92}, {196, 989, 92},
{197, 989, 91}, {198, 989, 81}, {199, 989, 91}, {200, 989, 92}, {201, 989, 92},
{202, 989, 91}, {203, 989, 81}, {204, 989, 91}, {205, 989, 90}, {206, 989, 90},
{207, 989, 89}, {208, 989, 80}, {209, 989, 89}, {210, 989, 90}, {211, 989, 90},
{212, 989, 89}, {213, 989, 80}, {214, 989, 89}, {215, 989, 88}, {216, 989, 88},
{217, 989, 87}, {218, 989, 79}, {219, 989, 87}, {220, 989, 88}, {221, 989, 88},
{222, 989, 87}, {223, 989, 79}, {224, 989, 87}, {225, 989, 86}, {226, 989, 86},
{227, 989, 85}, {228, 989, 78}, {229, 989, 85}, {230, 989, 86}, {231, 989, 86},
{232, 989, 85}, {233, 989, 78}, {234, 989, 85}, {235, 989, 84}, {236, 989, 84},
{237, 989, 83}, {238, 989, 77}, {239, 989, 83}, {240, 989, 84}, {241, 989, 84},
{242, 989, 83}, {243, 989, 77}, {244, 989, 83}, {245, 989, 82}, {246, 989, 82},
{247, 989, 81}, {248, 989, 76}, {249, 989, 81}, {250, 989, 82}, {251, 989, 82},
{252, 989, 81}, {253, 989, 76}, {254, 989, 81}, {255, 989, 80}, {256, 989, 80},
{257, 989, 79}, {258, 989, 75}, {259, 989, 79}, {260, 989, 80}, {261, 989, 80},
{262, 989, 79}, {263, 989, 75}, {264, 989, 79}, {265, 989, 78}, {266, 989, 78},
{267, 989, 77}, {268, 989, 74}, {269, 989, 77}, {270, 989, 78}, {271, 989, 78},
{272, 989, 77}, {273, 989, 74}, {274, 989, 77}, {275, 989, 76}, {276, 989, 76},
{277, 989, 75}, {278, 989, 73}, {279, 989, 75}, {280, 989, 76}, {281, 989, 76},
{282, 989, 75}, {283, 989, 73}, {284, 989, 75}, {285, 989, 74}, {286, 989, 74},
{287, 989, 73}, {288, 989, 72}, {289, 989, 73}, {290, 989, 74}, {291, 989, 74},
{292, 989, 73}, {293, 989, 72}, {294, 989, 73}, {295, 989, 72}, {296, 989, 72},
{297, 989, 68}, {298, 989, 71}, {299, 989, 71}, {300, 989, 72}, {301, 989, 72},
{302, 989, 68}, {303, 989, 71}, {304, 989, 71}, {305, 989, 71}, {306, 989, 70},
{307, 989, 67}, {308, 989, 70}, {309, 989, 70}, {310, 989, 71}, {311, 989, 70},
{312, 989, 67}, {313, 989, 70}, {314, 989, 70}, {315, 989, 69}, {316, 989, 69},
{317, 989, 66}, {318, 989, 68}, {319, 989, 69}, {320, 989, 69}, {321, 989, 69},
{322, 989, 66}, {323, 989, 68}, {324, 989, 69}, {325, 989, 68}, {326, 989, 68},
{327, 989, 65}, {328, 989, 67}, {329, 989, 67}, {330, 989, 68}, {331, 989, 68},
{332, 989, 65}, {333, 989, 67}, {334, 989, 67}, {335, 989, 67}, {336, 989, 66},
{337, 989, 64}, {338, 989, 66}, {339, 989, 66}, {340, 989, 67}, {341, 989, 66},
{342, 989, 64}, {343, 989, 66}, {344, 989, 66}, {345, 989, 65}, {346, 989, 65},
{347, 989, 63}, {348, 989, 64}, {349, 989, 65}, {350, 989, 65}, {351, 989, 65},
{352, 989, 63}, {353, 989, 64}, {354, 989, 65}, {355, 989, 64}, {356, 989, 64},
{357, 989, 62}, {358, 989, 63}, {359, 989, 63}, {360, 989, 64}, {361, 989, 64},
{362, 989, 62}, {363, 989, 63}, {364, 989, 63}, {365, 989, 63}, {366, 989, 62},
{367, 989, 61}, {368, 989, 62}, {369, 989, 62}, {370, 989, 63}, {371, 989, 62},
{372, 989, 61}, {373, 989, 62}, {374, 989, 62}, {375, 989, 61}, {376, 989, 61},
{377, 989, 60}, {378, 989, 60}, {379, 989, 61}, {380, 989, 61}, {381, 989, 61},
{382, 989, 60}, {383, 989, 60}, {384, 989, 61}, {385, 989, 60}, {386, 989, 60},
{387, 989, 59}, {388, 989, 59}, {389, 989, 59}, {390, 989, 60}, {391, 989, 60},
{392, 989, 59}, {393, 989, 59}, {394, 989, 59}, {395, 989, 59}, {396, 1000001, 121413}, {397, 989, 58}, {398, 1000001,
121416}, {399, 989, 58}, {400, 989, 59}, {401, 1000001, 121413}, {402, 989,
58}, {403, 1000001, 121416}, {404, 989, 58}, {405, 1000001, 121415}, {406,
1000001, 121412}, {407, 989, 57}, {408, 989, 57}, {409, 1000001, 121414}, {410,
1000001, 121415}, {411, 1000001, 121412}, {412, 989, 57}, {413, 989, 57}, {414,
1000001, 121414}, {415, 989, 56}, {416, 1000001, 121411}, {417, 1000001,
121413}, {418, 989, 56}, {419, 989, 55},
{420, 989, 56}, {421, 1000001, 121411}, {422, 1000001, 121413}, {423, 989, 56},
{424, 989, 55}, {425, 989, 55}, {426, 1000001, 121410}, {427, 989, 54}, {428,
1000001, 121412}, {429, 989, 54}, {430,
989, 55}, {431, 1000001, 121410}, {432, 989, 54}, {433, 1000001, 121412}, {434,
989, 54}, {435, 1000001, 121411}, {436, 1000001, 121409}, {437, 989, 53}, {438,
989, 53}, {439, 1000001, 121410}, {440, 1000001, 121411}, {441, 1000001,
121409}, {442, 989, 53}, {443, 989, 53}, {444, 1000001, 121410}, {445, 989,
52}, {446, 1000001, 121408}, {447, 1000001, 121409}, {448, 989, 52}, {449,
989, 51}, {450, 989, 52}, {451, 1000001,
121408}, {452, 1000001, 121409}, {453, 989, 52}, {454, 989, 51}, {455, 989,
51}, {456, 1000001, 121407}, {457, 989, 50}, {458, 1000001, 121408}, {459,
989, 50}, {460, 989, 51}, {461, 1000001,
121407}, {462, 989, 50}, {463, 1000001, 121408}, {464, 989, 50}, {465, 1000001,
121407}, {466, 1000001, 121406}, {467, 989, 49}, {468, 989, 49}, {469, 1000001,
121406}, {470, 1000001, 121407}, {471, 1000001, 121406}, {472, 989, 49}, {473,
989, 49}, {474, 1000001, 121406}, {475, 989, 48}, {476, 1000001, 121405}, {477,
1000001, 121405}, {478, 989, 48}, {479, 989,
47}, {480, 989, 48}, {481, 1000001, 121405}, {482, 1000001, 121405},
{483, 989, 48}, {484, 989, 47}, {485, 989, 47}, {486, 1000001, 121404}, {487,
989, 46}, {488, 1000001, 121404}, {489, 989,
46}, {490, 989, 47}, {491, 1000001, 121404}, {492, 989, 46}, {493,
1000001, 121404}, {494, 989, 46}, {495, 1000001, 121404}, {496, 1000001,
121403}, {497, 1000001, 121403}, {498, 989, 45}, {499, 989, 45}, {500, 1000001, 121404}, {501, 1000001,
121403}, {502, 1000001, 121403}, {503,
989, 45}, {504, 989, 45}, {505, 1000001, 121403}, {506, 989, 44}, {507,
1000001, 121402}, {508, 989, 44}, {509, 1000001, 121402}, {510, 1000001,
121403}, {511, 989, 44}, {512, 1000001, 121402}, {513, 989, 44}, {514, 1000001,
121402}, {515, 1000001, 121402}, {516, 989, 43}, {517, 989, 43}, {518, 1000001,
121401}, {519, 1000001, 121401}, {520, 1000001, 121402}, {521, 989, 43}, {522,
989, 43}, {523, 1000001, 121401}, {524, 1000001, 121401}, {525, 1000001,
121401}, {526, 1000001, 121400}, {527, 989, 42}, {528, 1000001, 121400}, {529,
989, 42}, {530, 1000001, 121401}, {531,
1000001, 121400}, {532, 989, 42}, {533, 1000001, 121400}, {534, 989, 42}, {535,
1000001, 121400}, {536, 1000001, 121399}, {537, 1000001, 121399}, {538, 989,
41}, {539, 989, 41}, {540, 1000001, 121400}, {541, 1000001, 121399}, {542,
1000001, 121399}, {543, 989, 41}, {544, 989, 41}, {545, 1000001, 121399}, {546,
989, 40}, {547, 1000001, 121398}, {548, 989, 40}, {549, 1000001, 121398}, {550,
1000001, 121399}, {551, 989, 40}, {552, 1000001, 121398}, {553, 989, 40}, {554,
1000001, 121398}, {555, 1000001, 121398}, {556, 989, 39}, {557, 989, 39}, {558,
1000001, 121397}, {559, 1000001, 121397}, {560, 1000001, 121398}, {561, 989,
39}, {562, 989, 39}, {563, 1000001, 121397}, {564, 1000001, 121397}, {565,
1000001, 121397}, {566, 1000001, 121396}, {567, 989, 38}, {568, 1000001,
121396}, {569, 989, 38}, {570, 1000001, 121397}, {571, 1000001, 121396}, {572,
989, 38}, {573, 1000001, 121396}, {574, 989, 38}, {575, 1000001, 121396}, {576,
1000001, 121395}, {577, 1000001, 121395}, {578, 989, 37}, {579, 989, 37}, {580,
1000001, 121396}, {581, 1000001, 121395}, {582, 1000001, 121395}, {583, 989,
37}, {584, 989, 37}, {585, 1000001, 121395}, {586, 989, 36}, {587, 1000001,
121394}, {588, 989, 36}, {589, 1000001, 121394}, {590, 1000001, 121395}, {591,
989, 36}, {592, 1000001, 121394}, {593, 989,
36}, {594, 1000001, 121394}, {595, 1000001, 121394}, {596, 1000001, 121394}, {597, 989, 35}, {598, 989, 35},
{599, 1000001, 121393}, {600, 1000001, 121394}, {601, 1000001, 121394}, {602,
989, 35}, {603, 989, 35}, {604, 1000001, 121393}, {605, 989, 34}, {606,
1000001, 121393}, {607, 989, 34}, {608, 1000001, 121393}, {609, 1000001,
121392}, {610, 989, 34}, {611, 1000001, 121393}, {612, 989, 34}, {613, 1000001, 121393}, {614, 1000001,
121392}, {615, 989, 33}, {616, 989, 33}, {617, 1000001, 121392}, {618, 1000001,
121392}, {619, 1000001, 121391}, {620, 989, 33}, {621, 989, 33}, {622, 1000001,
121392}, {623, 1000001, 121392}, {624, 1000001, 121391}, {625, 1000001,
121391}, {626, 989, 32}, {627, 1000001, 121391}, {628, 989, 32}, {629, 1000001,
121390}, {630, 1000001, 121391}, {631, 989, 32}, {632, 1000001, 121391}, {633,
989, 32}, {634, 1000001, 121390}, {635, 1000001, 121390}, {636, 1000001,
121390}, {637, 989, 31}, {638, 989, 31}, {639, 1000001, 121389}, {640, 1000001,
121390}, {641, 1000001, 121390}, {642, 989, 31}, {643, 989, 31}, {644, 1000001,
121389}, {645, 989, 30}, {646, 1000001, 121389}, {647, 989, 30}, {648, 1000001,
121389}, {649, 1000001, 121388}, {650, 989, 30}, {651, 1000001, 121389}, {652,
989, 30}, {653, 1000001, 121389}, {654,
1000001, 121388}, {655, 989, 29}, {656, 989, 29}, {657, 1000001, 121388}, {658,
1000001, 121388}, {659, 1000001, 121387}, {660, 989, 29}, {661, 989, 29}, {662,
1000001, 121388}, {663, 1000001, 121388}, {664, 1000001, 121387}, {665,
1000001, 121387}, {666, 989, 28}, {667, 1000001, 121387}, {668, 989, 28}, {669,
1000001, 121386}, {670, 1000001, 121387}, {671, 989, 28}, {672, 1000001,
121387}, {673, 989, 28}, {674, 1000001, 121386}, {675, 1000001, 121386}, {676,
1000001, 121386}, {677, 989, 27}, {678, 989, 27}, {679, 1000001, 121385}, {680,
1000001, 121386}, {681, 1000001, 121386}, {682, 989, 27}, {683, 989, 27}, {684,
1000001, 121385}, {685, 989, 26}, {686, 1000001, 121385}, {687, 989, 26}, {688,
1000001, 121385}, {689, 1000001, 121384}, {690, 989, 26}, {691, 1000001,
121385}, {692, 989, 26}, {693, 1000001,
121385}, {694, 1000001, 121384}, {695, 991, 26}, {696, 989, 25}, {697, 989,
25}, {698, 1000001, 121384}, {699, 1000001, 121382}, {700, 991, 26}, {701, 989,
25}, {702, 989, 25}, {703, 1000001, 121384}, {704, 1000001, 121382}, {705,
1000001, 121383}, {706, 989, 24}, {707, 991, 25}, {708, 1000001, 121383}, {709,
989, 24}, {710, 1000001, 121383}, {711, 989, 24}, {712, 991, 25}, {713,
1000001, 121383}, {714, 989, 24}, {715, 1000001, 121381}, {716, 991, 24}, {717,
1000001, 121382}, {718, 1000001, 121382}, {719, 989, 23}, {720, 1000001,
121381}, {721, 991, 24}, {722, 1000001,
121382}, {723, 1000001, 121382}, {724, 989, 23}, {725, 989, 23}, {726, 1000001,
121381}, {727, 1000001, 121380}, {728, 1000001, 121381}, {729, 991, 23}, {730,
989, 23}, {731, 1000001, 121381}, {732, 1000001, 121380}, {733, 1000001,
121381}, {734, 991, 23}, {735, 989, 22}, {736, 1000001, 121379}, {737, 989,
22}, {738, 1000001, 121380}, {739, 1000001, 121380}, {740, 989, 22}, {741,
1000001, 121379}, {742, 989, 22}, {743, 1000001, 121380}, {744, 1000001, 121380},
{745, 991, 22}, {746, 989, 21}, {747, 989, 21}, {748, 1000001, 121379}, {749,
1000001, 121378}, {750, 991, 22}, {751, 989, 21}, {752, 989, 21}, {753,
1000001, 121379}, {754, 1000001, 121378}, {755, 1000001, 121379}, {756, 989,
20}, {757, 991, 21}, {758, 1000001, 121378}, {759, 989, 20}, {760, 1000001,
121379}, {761, 989, 20}, {762, 991, 21}, {763, 1000001, 121378}, {764, 989,
20}, {765, 1000001, 121377}, {766, 991, 20}, {767, 1000001, 121378}, {768,
1000001, 121377}, {769, 989, 19}, {770, 1000001, 121377}, {771, 991, 20}, {772, 1000001, 121378}, {773, 1000001,
121377}, {774, 989, 19}, {775, 989, 19}, {776, 1000001, 121377}, {777, 1000001,
121376}, {778, 1000001, 121376}, {779, 991, 19}, {780, 989, 19}, {781, 1000001,
121377}, {782, 1000001, 121376}, {783, 1000001, 121376}, {784, 991, 19}, {785,
989, 18}, {786, 1000001, 121375}, {787, 989, 18}, {788, 1000001, 121375}, {789,
1000001, 121376}, {790, 989, 18}, {791, 1000001, 121375}, {792, 989, 18}, {793,
1000001, 121375}, {794, 1000001, 121376}, {795, 991, 18}, {796, 991, 18}, {797,
989, 17}, {798, 989, 15}, {799, 1000001, 121374}, {800, 991, 18}, {801, 991,
18}, {802, 989, 17}, {803, 989, 15}, {804, 1000001, 121374}, {805, 991, 17},
{806, 1000001, 121375}, {807, 989, 16}, {808, 991, 17}, {809, 989, 14}, {810,
991, 17}, {811, 1000001, 121375}, {812, 989, 16}, {813, 991, 17}, {814, 989,
14}, {815, 1000001, 121374}, {816, 1000001, 121373}, {817, 989, 15}, {818, 991, 16}, {819, 991, 16}, {820,
1000001, 121374}, {821, 1000001, 121373}, {822, 989, 15}, {823, 991, 16}, {824,
991, 16}, {825, 1000001, 121372}, {826, 989, 13}, {827, 989, 14}, {828,
1000001, 121373}, {829, 991, 15}, {830, 1000001, 121372}, {831, 989, 13}, {832, 989, 14}, {833, 1000001, 121373},
{834, 991, 15}, {835, 989, 12}, {836, 991, 15}, {837, 989, 13}, {838, 1000001,
121371}, {839, 1000001, 121372}, {840, 989, 12}, {841, 991, 15}, {842, 989,
13}, {843, 1000001, 121371}, {844, 1000001, 121372}, {845, 991, 14}, {846, 991,
14}, {847, 989, 12}, {848, 989, 11}, {849, 1000001, 121370}, {850, 991, 14},
{851, 991, 14}, {852, 989, 12}, {853, 989, 11}, {854, 1000001, 121370}, {855,
991, 13}, {856, 1000001, 121371}, {857, 989, 11}, {858, 991, 13}, {859, 989,
10}, {860, 991, 13}, {861, 1000001, 121371}, {862, 989, 11}, {863, 991, 13},
{864, 989, 10}, {865, 1000001, 121370}, {866, 1000001, 121369}, {867, 989, 10},
{868, 991, 12}, {869, 991, 12}, {870, 1000001, 121370}, {871, 1000001, 121369},
{872, 989, 10}, {873, 991, 12}, {874, 991, 12}, {875, 1000001, 121368}, {876,
989, 9}, {877, 989, 9}, {878, 1000001, 121369}, {879, 991, 11}, {880, 1000001,
121368}, {881, 989, 9}, {882, 989, 9}, {883, 1000001, 121369}, {884, 991, 11},
{885, 989, 8}, {886, 991, 11}, {887, 989, 8}, {888, 1000001, 121367}, {889,
1000001, 121368}, {890, 989, 8}, {891, 991, 11}, {892, 989, 8}, {893, 1000001,
121367}, {894, 1000001, 121368}, {895, 987, 7}, {896, 991, 10}, {897, 1000001,
121267}, {898, 989, 7}, {899, 993, 7}, {900, 987, 7}, {901, 991, 10}, {902,
1000001, 121267}, {903, 989, 7}, {904, 993, 7}, {905, 1000001, 121366}, {906,
991, 9}, {907, 1000001, 121367}, {908, 1000001, 121266}, {909, 987, 6}, {910,
1000001, 121366}, {911, 991, 9}, {912, 1000001, 121367}, {913, 1000001,
121266}, {914, 987, 6}, {915, 989, 6}, {916, 991, 8}, {917, 993, 6}, {918, 1000001, 121366}, {919,
1000001, 121365}, {920, 989, 6}, {921,
991, 8}, {922, 993, 6}, {923, 1000001, 121366}, {924, 1000001, 121365}, {925,
1000001, 121265}, {926, 991, 7}, {927, 987, 5}, {928, 993, 5}, {929, 989, 5},
{930, 1000001, 121265}, {931, 991, 7},
{932, 987, 5}, {933, 993, 5}, {934, 989, 5}, {935, 1000001, 121365}, {936, 991,
6}, {937, 1000001, 121364}, {938, 987, 4}, {939, 1000001, 121264}, {940, 1000001, 121365},
{941, 991, 6}, {942, 1000001, 121364}, {943, 987, 4}, {944, 1000001, 121264},
{945, 993, 4}, {946, 991, 5}, {947, 989, 4}, {948, 1000001, 121363}, {949,
1000001, 121364}, {950, 993, 4}, {951, 991, 5}, {952, 989, 4}, {953, 1000001, 121363}, {954, 1000001, 121364},
{955, 987, 3}, {956, 991, 4}, {957,
1000001, 121263}, {958, 989, 3}, {959, 993, 3}, {960, 987, 3}, {961, 991, 4},
{962, 1000001, 121263}, {963, 989, 3}, {964,
993, 3}, {965, 1000001, 121362}, {966, 991, 3}, {967, 1000001, 121363},
{968, 1000001, 121262}, {969, 987, 2}, {970, 1000001, 121362}, {971, 991, 3},
{972, 1000001, 121363}, {973, 1000001, 121262}, {974, 987, 2}, {975, 989, 2},
{976, 991, 2}, {977, 993, 2}, {978, 1000001, 121362}, {979, 1000001, 121361},
{980, 989, 2}, {981, 991, 2}, {982, 993, 2}, {983, 1000001, 121362}, {984,
1000001, 121361}, {985, 1000001, 121261}, {986, 986, 1}, {987, 987, 1}, {988,
988, 1}, {989, 989, 1}, {990, 1000001, 121261}, {991, 991, 1}, {992, 992, 1}, {993, 993, 1}, {994, 994,
1}, {995, 1000001, 121361}, {996, 1000001, 121360}, {997, 1000001, 121360},
{998, 1000001, 121359}, {999, 1000001, 121260}, {1000, 1000001, 121361}, {1001,
1000001, 121360}, {1002, 1000001, 121360}, {1003, 1000001, 121359}, {1004,
1000001, 121260}, {1005, 1000001, 121359}, {1006, 1000001, 121358}, {1007,
1000001, 121358}, {1008, 1000001, 121357}, {1009, 1000001, 121259}, {1010,
1000001, 121359}, {1011, 1000001, 121358}, {1012, 1000001, 121358}, {1013,
1000001, 121357}, {1014, 1000001, 121259}, {1015, 1000001, 121357}, {1016,
1000001, 121356}, {1017, 1000001, 121356}, {1018, 1000001, 121355}, {1019,
1000001, 121258}, {1020, 1000001, 121357}, {1021, 1000001, 121356}, {1022,
1000001, 121356}, {1023, 1000001, 121355}, {1024, 1000001, 121258}, {1025,
1000001, 121355}, {1026, 1000001, 121354}, {1027, 1000001, 121354}, {1028,
1000001, 121353}, {1029, 1000001, 121257}, {1030, 1000001, 121355}, {1031,
1000001, 121354}, {1032, 1000001, 121354}, {1033, 1000001, 121353}, {1034,
1000001, 121257}, {1035, 1000001, 121353}, {1036, 1000001, 121352}, {1037,
1000001, 121352}, {1038, 1000001, 121351}, {1039, 1000001, 121256}, {1040,
1000001, 121353}, {1041, 1000001, 121352}, {1042, 1000001, 121352}, {1043,
1000001, 121351}, {1044, 1000001, 121256}, {1045, 1000001, 121351}, {1046,
1000001, 121350}, {1047, 1000001, 121350}, {1048, 1000001, 121349}, {1049,
1000001, 121255}, {1050, 1000001, 121351}, {1051, 1000001, 121350}, {1052,
1000001, 121350}, {1053, 1000001, 121349}, {1054, 1000001, 121255}, {1055,
1000001, 121349}, {1056, 1000001, 121348}, {1057, 1000001, 121348}, {1058,
1000001, 121347}, {1059, 1000001, 121254}, {1060, 1000001, 121349}, {1061,
1000001, 121348}, {1062, 1000001, 121348}, {1063, 1000001, 121347}, {1064,
1000001, 121254}, {1065, 1000001, 121347}, {1066, 1000001, 121346}, {1067,
1000001, 121346}, {1068, 1000001, 121345}, {1069, 1000001, 121253}, {1070,
1000001, 121347}, {1071, 1000001, 121346}, {1072, 1000001, 121346}, {1073,
1000001, 121345}, {1074, 1000001, 121253}, {1075, 1000001, 121345}, {1076,
1000001, 121344}, {1077, 1000001, 121344}, {1078, 1000001, 121343}, {1079,
1000001, 121252}, {1080, 1000001, 121345}, {1081, 1000001, 121344}, {1082,
1000001, 121344}, {1083, 1000001, 121343}, {1084, 1000001, 121252}, {1085,
1000001, 121343}, {1086, 1000001, 121342}, {1087, 1000001, 121342}, {1088,
1000001, 121341}, {1089, 1000001, 121251}, {1090, 1000001, 121343}, {1091,
1000001, 121342}, {1092, 1000001, 121342}, {1093, 1000001, 121341}, {1094,
1000001, 121251}, {1095, 1000001, 121341}, {1096, 1000001, 121340}, {1097,
1000001, 121340}, {1098, 1000001, 121339}, {1099, 1000001, 121250}, {1100,
1000001, 121341}, {1101, 1000001, 121340}, {1102, 1000001, 121340}, {1103, 1000001,
121339}, {1104, 1000001, 121250}, {1105, 1000001, 121339}, {1106, 1000001,
121338}, {1107, 1000001, 121338}, {1108, 1000001, 121337}, {1109, 1000001,
121249}, {1110, 1000001, 121339}, {1111, 1000001, 121338}, {1112, 1000001,
121338}, {1113, 1000001, 121337}, {1114, 1000001, 121249}, {1115, 1000001,
121337}, {1116, 1000001, 121336}, {1117, 1000001, 121336}, {1118, 1000001,
121335}, {1119, 1000001, 121248}, {1120, 1000001, 121337}, {1121, 1000001,
121336}, {1122, 1000001, 121336}, {1123, 1000001, 121335}, {1124, 1000001,
121248}, {1125, 1000001, 121335}, {1126, 1000001, 121334}, {1127, 1000001,
121334}, {1128, 1000001, 121333}, {1129, 1000001, 121247}, {1130, 1000001,
121335}, {1131, 1000001, 121334}, {1132, 1000001, 121334}, {1133, 1000001,
121333}, {1134, 1000001, 121247}, {1135, 1000001, 121333}, {1136, 1000001,
121332}, {1137, 1000001, 121332}, {1138, 1000001, 121331}, {1139, 1000001,
121246}, {1140, 1000001, 121333}, {1141, 1000001, 121332}, {1142, 1000001,
121332}, {1143, 1000001, 121331}, {1144, 1000001, 121246}, {1145, 1000001,
121331}, {1146, 1000001, 121330}, {1147, 1000001, 121330}, {1148, 1000001,
121329}, {1149, 1000001, 121245}, {1150, 1000001, 121331}, {1151, 1000001,
121330}, {1152, 1000001, 121330}, {1153, 1000001, 121329}, {1154, 1000001,
121245}, {1155, 1000001, 121329}, {1156, 1000001, 121328}, {1157, 1000001,
121328}, {1158, 1000001, 121327}, {1159, 1000001, 121244}, {1160, 1000001,
121329}, {1161, 1000001, 121328}, {1162, 1000001, 121328}, {1163, 1000001,
121327}, {1164, 1000001, 121244}, {1165, 1000001, 121327}, {1166, 1000001,
121326}, {1167, 1000001, 121326}, {1168, 1000001, 121325}, {1169, 1000001,
121243}, {1170, 1000001, 121327}, {1171, 1000001, 121326}, {1172, 1000001,
121326}, {1173, 1000001, 121325}, {1174, 1000001, 121243}, {1175, 1000001,
121325}, {1176, 1000001, 121324}, {1177, 1000001, 121324}, {1178, 1000001,
121323}, {1179, 1000001, 121242}, {1180, 1000001, 121325}, {1181, 1000001,
121324}, {1182, 1000001, 121324}, {1183, 1000001, 121323}, {1184, 1000001,
121242}, {1185, 1000001, 121323}, {1186, 1000001, 121322}, {1187, 1000001,
121322}, {1188, 1000001, 121321}, {1189, 1000001, 121241}, {1190, 1000001,
121323}, {1191, 1000001, 121322}, {1192, 1000001, 121322}, {1193, 1000001,
121321}, {1194, 1000001, 121241}, {1195, 1000001, 121321}, {1196, 1000001,
121320}, {1197, 1000001, 121320}, {1198, 1000001, 121319}, {1199, 1000001,
121240}, {1200, 1000001, 121321}, {1201, 1000001, 121320}, {1202, 1000001,
121320}, {1203, 1000001, 121319}, {1204, 1000001, 121240}, {1205, 1000001, 121319},
{1206, 1000001, 121318}, {1207, 1000001, 121318}, {1208, 1000001, 121317},
{1209, 1000001, 121239}, {1210, 1000001, 121319}, {1211, 1000001, 121318},
{1212, 1000001, 121318}, {1213, 1000001, 121317}, {1214, 1000001, 121239},
{1215, 1000001, 121317}, {1216, 1000001, 121316}, {1217, 1000001, 121316},
{1218, 1000001, 121315}, {1219, 1000001, 121238}, {1220, 1000001, 121317},
{1221, 1000001, 121316}, {1222, 1000001, 121316}, {1223, 1000001, 121315},
{1224, 1000001, 121238}, {1225, 1000001, 121315}, {1226, 1000001, 121314},
{1227, 1000001, 121314}, {1228, 1000001, 121313}, {1229, 1000001, 121237},
{1230, 1000001, 121315}, {1231, 1000001, 121314}, {1232, 1000001, 121314},
{1233, 1000001, 121313}, {1234, 1000001, 121237}, {1235, 1000001, 121313},
{1236, 1000001, 121312}, {1237, 1000001, 121312}, {1238, 1000001, 121311},
{1239, 1000001, 121236}, {1240, 1000001, 121313}, {1241, 1000001, 121312},
{1242, 1000001, 121312}, {1243, 1000001, 121311}, {1244, 1000001, 121236},
{1245, 1000001, 121311}, {1246, 1000001, 121310}, {1247, 1000001, 121310},
{1248, 1000001, 121309}, {1249, 1000001, 121235}, {1250, 1000001, 121311},
{1251, 1000001, 121310}, {1252, 1000001, 121310}, {1253, 1000001, 121309},
{1254, 1000001, 121235}, {1255, 1000001, 121309}, {1256, 1000001, 121308}, {1257,
1000001, 121308}, {1258, 1000001, 121307}, {1259, 1000001, 121234}, {1260,
1000001, 121309}, {1261, 1000001, 121308}, {1262, 1000001, 121308}, {1263,
1000001, 121307}, {1264, 1000001, 121234}, {1265, 1000001, 121307}, {1266,
1000001, 121306}, {1267, 1000001, 121306}, {1268, 1000001, 121305}, {1269,
1000001, 121233}, {1270, 1000001, 121307}, {1271, 1000001, 121306}, {1272,
1000001, 121306}, {1273, 1000001, 121305}, {1274, 1000001, 121233}, {1275,
1000001, 121305}, {1276, 1000001, 121304}, {1277, 1000001, 121304}, {1278,
1000001, 121303}, {1279, 1000001, 121232}, {1280, 1000001, 121305}, {1281,
1000001, 121304}, {1282, 1000001, 121304}, {1283, 1000001, 121303}, {1284,
1000001, 121232}, {1285, 1000001, 121303}, {1286, 1000001, 121302}, {1287,
1000001, 121302}, {1288, 1000001, 121301}, {1289, 1000001, 121231}, {1290,
1000001, 121303}, {1291, 1000001, 121302}, {1292, 1000001, 121302}, {1293,
1000001, 121301}, {1294, 1000001, 121231}, {1295, 1000001, 121301}, {1296,
1000001, 121300}, {1297, 1000001, 121300}, {1298, 1000001, 121299}, {1299,
1000001, 121230}, {1300, 1000001, 121301}, {1301, 1000001, 121300}, {1302,
1000001, 121300}, {1303, 1000001, 121299}, {1304, 1000001, 121230}, {1305,
1000001, 121299}, {1306, 1000001, 121298}, {1307, 1000001, 121298}, {1308,
1000001, 121297}, {1309, 1000001, 121229}, {1310, 1000001, 121299}, {1311,
1000001, 121298}, {1312, 1000001, 121298}, {1313, 1000001, 121297}, {1314,
1000001, 121229}, {1315, 1000001, 121297}, {1316, 1000001, 121296}, {1317,
1000001, 121296}, {1318, 1000001, 121295}, {1319, 1000001, 121228}, {1320,
1000001, 121297}, {1321, 1000001, 121296}, {1322, 1000001, 121296}, {1323,
1000001, 121295}, {1324, 1000001, 121228}, {1325, 1000001, 121295}, {1326,
1000001, 121294}, {1327, 1000001, 121294}, {1328, 1000001, 121293}, {1329,
1000001, 121227}, {1330, 1000001, 121295}, {1331, 1000001, 121294}, {1332,
1000001, 121294}, {1333, 1000001, 121293}, {1334, 1000001, 121227}, {1335,
1000001, 121293}, {1336, 1000001, 121292}, {1337, 1000001, 121292}, {1338,
1000001, 121291}, {1339, 1000001, 121226}, {1340, 1000001, 121293}, {1341,
1000001, 121292}, {1342, 1000001, 121292}, {1343, 1000001, 121291}, {1344,
1000001, 121226}, {1345, 1000001, 121291}, {1346, 1000001, 121290}, {1347,
1000001, 121290}, {1348, 1000001, 121289}, {1349, 1000001, 121225}, {1350,
1000001, 121291}, {1351, 1000001, 121290}, {1352, 1000001, 121290}, {1353,
1000001, 121289}, {1354, 1000001, 121225}, {1355, 1000001, 121289}, {1356,
1000001, 121288}, {1357, 1000001, 121288}, {1358, 1000001, 121287}, {1359, 1000001,
121224}, {1360, 1000001, 121289}, {1361, 1000001, 121288}, {1362, 1000001,
121288}, {1363, 1000001, 121287}, {1364, 1000001, 121224}, {1365, 1000001,
121287}, {1366, 1000001, 121286}, {1367, 1000001, 121286}, {1368, 1000001,
121285}, {1369, 1000001, 121223}, {1370, 1000001, 121287}, {1371, 1000001,
121286}, {1372, 1000001, 121286}, {1373, 1000001, 121285}, {1374, 1000001,
121223}, {1375, 1000001, 121285}, {1376, 1000001, 121284}, {1377, 1000001,
121284}, {1378, 1000001, 121283}, {1379, 1000001, 121222}, {1380, 1000001,
121285}, {1381, 1000001, 121284}, {1382, 1000001, 121284}, {1383, 1000001,
121283}, {1384, 1000001, 121222}, {1385, 1000001, 121283}, {1386, 1000001,
121282}, {1387, 1000001, 121282}, {1388, 1000001, 121281}, {1389, 1000001,
121221}, {1390, 1000001, 121283}, {1391, 1000001, 121282}, {1392, 1000001,
121282}, {1393, 1000001, 121281}, {1394, 1000001, 121221}, {1395, 1000001,
121281}, {1396, 1000001, 121280}, {1397, 1000001, 121280}, {1398, 1000001,
121279}, {1399, 1000001, 121220}, {1400, 1000001, 121281}, {1401, 1000001,
121280}, {1402, 1000001, 121280}, {1403, 1000001, 121279}, {1404, 1000001,
121220}, {1405, 1000001, 121279}, {1406, 1000001, 121278}, {1407, 1000001,
121278}, {1408, 1000001, 121277}, {1409, 1000001, 121219}, {1410, 1000001,
121279}, {1411, 1000001, 121278}, {1412, 1000001, 121278}, {1413, 1000001,
121277}, {1414, 1000001, 121219}, {1415, 1000001, 121277}, {1416, 1000001,
121276}, {1417, 1000001, 121276}, {1418, 1000001, 121275}, {1419, 1000001,
121218}, {1420, 1000001, 121277}, {1421, 1000001, 121276}, {1422, 1000001,
121276}, {1423, 1000001, 121275}, {1424, 1000001, 121218}, {1425, 1000001,
121275}, {1426, 1000001, 121274}, {1427, 1000001, 121274}, {1428, 1000001,
121273}, {1429, 1000001, 121217}, {1430, 1000001, 121275}, {1431, 1000001,
121274}, {1432, 1000001, 121274}, {1433, 1000001, 121273}, {1434, 1000001,
121217}, {1435, 1000001, 121273}, {1436, 1000001, 121272}, {1437, 1000001,
121272}, {1438, 1000001, 121271}, {1439, 1000001, 121216}, {1440, 1000001,
121273}, {1441, 1000001, 121272}, {1442, 1000001, 121272}, {1443, 1000001,
121271}, {1444, 1000001, 121216}, {1445, 1000001, 121271}, {1446, 1000001,
121270}, {1447, 1000001, 121270}, {1448, 1000001, 121269}, {1449, 1000001,
121215}, {1450, 1000001, 121271}, {1451, 1000001, 121270}, {1452, 1000001,
121270}, {1453, 1000001, 121269}, {1454, 1000001, 121215}, {1455, 1000001,
121269}, {1456, 1000001, 121268}, {1457, 1000001, 121268}, {1458, 1000001,
121267}, {1459, 1000001, 121214}, {1460, 1000001, 121269}, {1461, 1000001, 121268},
{1462, 1000001, 121268}, {1463, 1000001, 121267}, {1464, 1000001, 121214},
{1465, 1000001, 121267}, {1466, 1000001, 121266}, {1467, 1000001, 121266},
{1468, 1000001, 121265}, {1469, 1000001, 121213}, {1470, 1000001, 121267},
{1471, 1000001, 121266}, {1472, 1000001, 121266}, {1473, 1000001, 121265},
{1474, 1000001, 121213}, {1475, 1000001, 121265}, {1476, 1000001, 121264},
{1477, 1000001, 121264}, {1478, 1000001, 121263}, {1479, 1000001, 121212},
{1480, 1000001, 121265}, {1481, 1000001, 121264}, {1482, 1000001, 121264},
{1483, 1000001, 121263}, {1484, 1000001, 121212}, {1485, 1000001, 121263},
{1486, 1000001, 121262}, {1487, 1000001, 121262}, {1488, 1000001, 121261},
{1489, 1000001, 121211}, {1490, 1000001, 121263}, {1491, 1000001, 121262},
{1492, 1000001, 121262}, {1493, 1000001, 121261}, {1494, 1000001, 121211},
{1495, 1000001, 121261}, {1496, 1000001, 121260}, {1497, 1000001, 121260},
{1498, 1000001, 121259}, {1499, 1000001, 121210}, {1500, 1000001, 121261},
{1501, 1000001, 121260}, {1502, 1000001, 121260}, {1503, 1000001, 121259},
{1504, 1000001, 121210}, {1505, 1000001, 121259}, {1506, 1000001, 121258},
{1507, 1000001, 121258}, {1508, 1000001, 121257}, {1509, 1000001, 121209},
{1510, 1000001, 121259}, {1511, 1000001, 121258}, {1512, 1000001, 121258}, {1513,
1000001, 121257}, {1514, 1000001, 121209}, {1515, 1000001, 121257}, {1516,
1000001, 121256}, {1517, 1000001, 121256}, {1518, 1000001, 121255}, {1519,
1000001, 121208}, {1520, 1000001, 121257}, {1521, 1000001, 121256}, {1522,
1000001, 121256}, {1523, 1000001, 121255}, {1524, 1000001, 121208}, {1525,
1000001, 121255}, {1526, 1000001, 121254}, {1527, 1000001, 121254}, {1528,
1000001, 121253}, {1529, 1000001, 121207}, {1530, 1000001, 121255}, {1531,
1000001, 121254}, {1532, 1000001, 121254}, {1533, 1000001, 121253}, {1534,
1000001, 121207}, {1535, 1000001, 121253}, {1536, 1000001, 121252}, {1537,
1000001, 121252}, {1538, 1000001, 121251}, {1539, 1000001, 121206}, {1540,
1000001, 121253}, {1541, 1000001, 121252}, {1542, 1000001, 121252}, {1543,
1000001, 121251}, {1544, 1000001, 121206}, {1545, 1000001, 121251}, {1546,
1000001, 121250}, {1547, 1000001, 121250}, {1548, 1000001, 121249}, {1549,
1000001, 121205}, {1550, 1000001, 121251}, {1551, 1000001, 121250}, {1552,
1000001, 121250}, {1553, 1000001, 121249}, {1554, 1000001, 121205}, {1555,
1000001, 121249}, {1556, 1000001, 121248}, {1557, 1000001, 121248}, {1558,
1000001, 121247}, {1559, 1000001, 121204}, {1560, 1000001, 121249}, {1561,
1000001, 121248}, {1562, 1000001, 121248}, {1563, 1000001, 121247}, {1564,
1000001, 121204}, {1565, 1000001, 121247}, {1566, 1000001, 121246}, {1567,
1000001, 121246}, {1568, 1000001, 121245}, {1569, 1000001, 121203}, {1570,
1000001, 121247}, {1571, 1000001, 121246}, {1572, 1000001, 121246}, {1573,
1000001, 121245}, {1574, 1000001, 121203}, {1575, 1000001, 121245}, {1576,
1000001, 121244}, {1577, 1000001, 121244}, {1578, 1000001, 121243}, {1579,
1000001, 121202}, {1580, 1000001, 121245}, {1581, 1000001, 121244}, {1582,
1000001, 121244}, {1583, 1000001, 121243}, {1584, 1000001, 121202}, {1585,
1000001, 121243}, {1586, 1000001, 121242}, {1587, 1000001, 121242}, {1588,
1000001, 121241}, {1589, 1000001, 121201}, {1590, 1000001, 121243}, {1591,
1000001, 121242}, {1592, 1000001, 121242}, {1593, 1000001, 121241}, {1594,
1000001, 121201}, {1595, 1000001, 121241}, {1596, 1000001, 121240}, {1597,
1000001, 121240}, {1598, 1000001, 121239}, {1599, 1000001, 121200}, {1600,
1000001, 121241}, {1601, 1000001, 121240}, {1602, 1000001, 121240}, {1603,
1000001, 121239}, {1604, 1000001, 121200}, {1605, 1000001, 121239}, {1606,
1000001, 121238}, {1607, 1000001, 121238}, {1608, 1000001, 121237}, {1609,
1000001, 121199}, {1610, 1000001, 121239}, {1611, 1000001, 121238}, {1612,
1000001, 121238}, {1613, 1000001, 121237}, {1614, 1000001, 121199}, {1615, 1000001,
121237}, {1616, 1000001, 121236}, {1617, 1000001, 121236}, {1618, 1000001,
121235}, {1619, 1000001, 121198}, {1620, 1000001, 121237}, {1621, 1000001,
121236}, {1622, 1000001, 121236}, {1623, 1000001, 121235}, {1624, 1000001,
121198}, {1625, 1000001, 121235}, {1626, 1000001, 121234}, {1627, 1000001,
121234}, {1628, 1000001, 121233}, {1629, 1000001, 121197}, {1630, 1000001,
121235}, {1631, 1000001, 121234}, {1632, 1000001, 121234}, {1633, 1000001,
121233}, {1634, 1000001, 121197}, {1635, 1000001, 121233}, {1636, 1000001,
121232}, {1637, 1000001, 121232}, {1638, 1000001, 121231}, {1639, 1000001,
121196}, {1640, 1000001, 121233}, {1641, 1000001, 121232}, {1642, 1000001,
121232}, {1643, 1000001, 121231}, {1644, 1000001, 121196}, {1645, 1000001,
121231}, {1646, 1000001, 121230}, {1647, 1000001, 121230}, {1648, 1000001,
121229}, {1649, 1000001, 121195}, {1650, 1000001, 121231}, {1651, 1000001,
121230}, {1652, 1000001, 121230}, {1653, 1000001, 121229}, {1654, 1000001,
121195}, {1655, 1000001, 121229}, {1656, 1000001, 121228}, {1657, 1000001,
121228}, {1658, 1000001, 121227}, {1659, 1000001, 121194}, {1660, 1000001,
121229}, {1661, 1000001, 121228}, {1662, 1000001, 121228}, {1663, 1000001,
121227}, {1664, 1000001, 121194}, {1665, 1000001, 121227}, {1666, 1000001,
121226}, {1667, 1000001, 121226}, {1668, 1000001, 121225}, {1669, 1000001,
121193}, {1670, 1000001, 121227}, {1671, 1000001, 121226}, {1672, 1000001,
121226}, {1673, 1000001, 121225}, {1674, 1000001, 121193}, {1675, 1000001,
121225}, {1676, 1000001, 121224}, {1677, 1000001, 121224}, {1678, 1000001,
121223}, {1679, 1000001, 121192}, {1680, 1000001, 121225}, {1681, 1000001,
121224}, {1682, 1000001, 121224}, {1683, 1000001, 121223}, {1684, 1000001,
121192}, {1685, 1000001, 121223}, {1686, 1000001, 121222}, {1687, 1000001,
121222}, {1688, 1000001, 121221}, {1689, 1000001, 121191}, {1690, 1000001,
121223}, {1691, 1000001, 121222}, {1692, 1000001, 121222}, {1693, 1000001,
121221}, {1694, 1000001, 121191}, {1695, 1000001, 121221}, {1696, 1000001,
121220}, {1697, 1000001, 121220}, {1698, 1000001, 121219}, {1699, 1000001,
121190}, {1700, 1000001, 121221}, {1701, 1000001, 121220}, {1702, 1000001,
121220}, {1703, 1000001, 121219}, {1704, 1000001, 121190}, {1705, 1000001,
121219}, {1706, 1000001, 121218}, {1707, 1000001, 121218}, {1708, 1000001,
121217}, {1709, 1000001, 121189}, {1710, 1000001, 121219}, {1711, 1000001,
121218}, {1712, 1000001, 121218}, {1713, 1000001, 121217}, {1714, 1000001,
121189}, {1715, 1000001, 121217}, {1716, 1000001, 121216}, {1717, 1000001, 121216},
{1718, 1000001, 121215}, {1719, 1000001, 121188}, {1720, 1000001, 121217},
{1721, 1000001, 121216}, {1722, 1000001, 121216}, {1723, 1000001, 121215},
{1724, 1000001, 121188}, {1725, 1000001, 121215}, {1726, 1000001, 121214},
{1727, 1000001, 121214}, {1728, 1000001, 121213}, {1729, 1000001, 121187},
{1730, 1000001, 121215}, {1731, 1000001, 121214}, {1732, 1000001, 121214},
{1733, 1000001, 121213}, {1734, 1000001, 121187}, {1735, 1000001, 121213},
{1736, 1000001, 121212}, {1737, 1000001, 121212}, {1738, 1000001, 121211},
{1739, 1000001, 121186}, {1740, 1000001, 121213}, {1741, 1000001, 121212},
{1742, 1000001, 121212}, {1743, 1000001, 121211}, {1744, 1000001, 121186},
{1745, 1000001, 121211}, {1746, 1000001, 121210}, {1747, 1000001, 121210},
{1748, 1000001, 121209}, {1749, 1000001, 121185}, {1750, 1000001, 121211},
{1751, 1000001, 121210}, {1752, 1000001, 121210}, {1753, 1000001, 121209},
{1754, 1000001, 121185}, {1755, 1000001, 121209}, {1756, 1000001, 121208},
{1757, 1000001, 121208}, {1758, 1000001, 121207}, {1759, 1000001, 121184},
{1760, 1000001, 121209}, {1761, 1000001, 121208}, {1762, 1000001, 121208},
{1763, 1000001, 121207}, {1764, 1000001, 121184}, {1765, 1000001, 121207},
{1766, 1000001, 121206}, {1767, 1000001, 121206}, {1768, 1000001, 121205}, {1769,
1000001, 121183}, {1770, 1000001, 121207}, {1771, 1000001, 121206}, {1772,
1000001, 121206}, {1773, 1000001, 121205}, {1774, 1000001, 121183}, {1775,
1000001, 121205}, {1776, 1000001, 121204}, {1777, 1000001, 121204}, {1778,
1000001, 121203}, {1779, 1000001, 121182}, {1780, 1000001, 121205}, {1781,
1000001, 121204}, {1782, 1000001, 121204}, {1783, 1000001, 121203}, {1784,
1000001, 121182}, {1785, 1000001, 121203}, {1786, 1000001, 121202}, {1787,
1000001, 121202}, {1788, 1000001, 121201}, {1789, 1000001, 121181}, {1790,
1000001, 121203}, {1791, 1000001, 121202}, {1792, 1000001, 121202}, {1793,
1000001, 121201}, {1794, 1000001, 121181}, {1795, 1000001, 121201}, {1796,
1000001, 121200}, {1797, 1000001, 121200}, {1798, 1000001, 121199}, {1799,
1000001, 121180}, {1800, 1000001, 121201}, {1801, 1000001, 121200}, {1802,
1000001, 121200}, {1803, 1000001, 121199}, {1804, 1000001, 121180}, {1805,
1000001, 121199}, {1806, 1000001, 121198}, {1807, 1000001, 121198}, {1808,
1000001, 121197}, {1809, 1000001, 121179}, {1810, 1000001, 121199}, {1811,
1000001, 121198}, {1812, 1000001, 121198}, {1813, 1000001, 121197}, {1814,
1000001, 121179}, {1815, 1000001, 121197}, {1816, 1000001, 121196}, {1817,
1000001, 121196}, {1818, 1000001, 121195}, {1819, 1000001, 121178}, {1820,
1000001, 121197}, {1821, 1000001, 121196}, {1822, 1000001, 121196}, {1823,
1000001, 121195}, {1824, 1000001, 121178}, {1825, 1000001, 121195}, {1826,
1000001, 121194}, {1827, 1000001, 121194}, {1828, 1000001, 121193}, {1829,
1000001, 121177}, {1830, 1000001, 121195}, {1831, 1000001, 121194}, {1832,
1000001, 121194}, {1833, 1000001, 121193}, {1834, 1000001, 121177}, {1835,
1000001, 121193}, {1836, 1000001, 121192}, {1837, 1000001, 121192}, {1838,
1000001, 121191}, {1839, 1000001, 121176}, {1840, 1000001, 121193}, {1841,
1000001, 121192}, {1842, 1000001, 121192}, {1843, 1000001, 121191}, {1844,
1000001, 121176}, {1845, 1000001, 121191}, {1846, 1000001, 121190}, {1847,
1000001, 121190}, {1848, 1000001, 121189}, {1849, 1000001, 121175}, {1850,
1000001, 121191}, {1851, 1000001, 121190}, {1852, 1000001, 121190}, {1853,
1000001, 121189}, {1854, 1000001, 121175}, {1855, 1000001, 121189}, {1856,
1000001, 121188}, {1857, 1000001, 121188}, {1858, 1000001, 121187}, {1859,
1000001, 121174}, {1860, 1000001, 121189}, {1861, 1000001, 121188}, {1862,
1000001, 121188}, {1863, 1000001, 121187}, {1864, 1000001, 121174}, {1865,
1000001, 121187}, {1866, 1000001, 121186}, {1867, 1000001, 121186}, {1868,
1000001, 121185}, {1869, 1000001, 121173}, {1870, 1000001, 121187}, {1871, 1000001,
121186}, {1872, 1000001, 121186}, {1873, 1000001, 121185}, {1874, 1000001,
121173}, {1875, 1000001, 121185}, {1876, 1000001, 121184}, {1877, 1000001,
121184}, {1878, 1000001, 121183}, {1879, 1000001, 121172}, {1880, 1000001,
121185}, {1881, 1000001, 121184}, {1882, 1000001, 121184}, {1883, 1000001,
121183}, {1884, 1000001, 121172}, {1885, 1000001, 121183}, {1886, 1000001,
121182}, {1887, 1000001, 121182}, {1888, 1000001, 121181}, {1889, 1000001,
121171}, {1890, 1000001, 121183}, {1891, 1000001, 121182}, {1892, 1000001,
121182}, {1893, 1000001, 121181}, {1894, 1000001, 121171}, {1895, 1000001,
121181}, {1896, 1000001, 121180}, {1897, 1000001, 121180}, {1898, 1000001,
121179}, {1899, 1000001, 121170}, {1900, 1000001, 121181}, {1901, 1000001,
121180}, {1902, 1000001, 121180}, {1903, 1000001, 121179}, {1904, 1000001,
121170}, {1905, 1000001, 121179}, {1906, 1000001, 121178}, {1907, 1000001,
121178}, {1908, 1000001, 121177}, {1909, 1000001, 121169}, {1910, 1000001,
121179}, {1911, 1000001, 121178}, {1912, 1000001, 121178}, {1913, 1000001,
121177}, {1914, 1000001, 121169}, {1915, 1000001, 121177}, {1916, 1000001,
121176}, {1917, 1000001, 121176}, {1918, 1000001, 121175}, {1919, 1000001,
121168}, {1920, 1000001, 121177}, {1921, 1000001, 121176}, {1922, 1000001,
121176}, {1923, 1000001, 121175}, {1924, 1000001, 121168}, {1925, 1000001,
121175}, {1926, 1000001, 121174}, {1927, 1000001, 121174}, {1928, 1000001,
121173}, {1929, 1000001, 121167}, {1930, 1000001, 121175}, {1931, 1000001,
121174}, {1932, 1000001, 121174}, {1933, 1000001, 121173}, {1934, 1000001,
121167}, {1935, 1000001, 121173}, {1936, 1000001, 121172}, {1937, 1000001,
121172}, {1938, 1000001, 121171}, {1939, 1000001, 121166}, {1940, 1000001,
121173}, {1941, 1000001, 121172}, {1942, 1000001, 121172}, {1943, 1000001,
121171}, {1944, 1000001, 121166}, {1945, 1000001, 121171}, {1946, 1000001,
121170}, {1947, 1000001, 121170}, {1948, 1000001, 121169}, {1949, 1000001,
121165}, {1950, 1000001, 121171}, {1951, 1000001, 121170}, {1952, 1000001,
121170}, {1953, 1000001, 121169}, {1954, 1000001, 121165}, {1955, 1000001,
121169}, {1956, 1000001, 121168}, {1957, 1000001, 121168}, {1958, 1000001,
121167}, {1959, 1000001, 121164}, {1960, 1000001, 121169}, {1961, 1000001,
121168}, {1962, 1000001, 121168}, {1963, 1000001, 121167}, {1964, 1000001,
121164}, {1965, 1000001, 121167}, {1966, 1000001, 121166}, {1967, 1000001,
121166}, {1968, 1000001, 121165}, {1969, 1000001, 121163}, {1970, 1000001,
121167}, {1971, 1000001, 121166}, {1972, 1000001, 121166}, {1973, 1000001, 121165},
{1974, 1000001, 121163}, {1975, 1000001, 121165}, {1976, 1000001, 121164},
{1977, 1000001, 121164}, {1978, 1000001, 121163}, {1979, 1000001, 121162},
{1980, 1000001, 121165}, {1981, 1000001, 121164}, {1982, 1000001, 121164},
{1983, 1000001, 121163}, {1984, 1000001, 121162}, {1985, 1000001, 121163},
{1986, 1000001, 121162}, {1987, 1000001, 121162}, {1988, 1000001, 121161},
{1989, 1000001, 121161}, {1990, 1000001, 121163}, {1991, 1000001, 121162},
{1992, 1000001, 121162}, {1993, 1000001, 121161}, {1994, 1000001, 121161},
{1995, 1000001, 121161}, {1996, 1000001, 121161}, {1997, 1000001, 121160},
{1998, 1000001, 121060}, {1999, 1000001, 121160}, {2000, 1000001, 121161},
{2001, 1000001, 121161}, {2002, 1000001, 121160}, {2003, 1000001, 121060},
{2004, 1000001, 121160}, {2005, 1000001, 121159}, {2006, 1000001, 121159},
{2007, 1000001, 121158}, {2008, 1000001, 121059}, {2009, 1000001, 121158},
{2010, 1000001, 121159}, {2011, 1000001, 121159}, {2012, 1000001, 121158},
{2013, 1000001, 121059}, {2014, 1000001, 121158}, {2015, 1000001, 121157},
{2016, 1000001, 121157}, {2017, 1000001, 121156}, {2018, 1000001, 121058},
{2019, 1000001, 121156}, {2020, 1000001, 121157}, {2021, 1000001, 121157},
{2022, 1000001, 121156}, {2023, 1000001, 121058}, {2024, 1000001, 121156}, {2025,
1000001, 121155}, {2026, 1000001, 121155}, {2027, 1000001, 121154}, {2028,
1000001, 121057}, {2029, 1000001, 121154}, {2030, 1000001, 121155}, {2031,
1000001, 121155}, {2032, 1000001, 121154}, {2033, 1000001, 121057}, {2034,
1000001, 121154}, {2035, 1000001, 121153}, {2036, 1000001, 121153}, {2037,
1000001, 121152}, {2038, 1000001, 121056}, {2039, 1000001, 121152}, {2040,
1000001, 121153}, {2041, 1000001, 121153}, {2042, 1000001, 121152}, {2043,
1000001, 121056}, {2044, 1000001, 121152}, {2045, 1000001, 121151}, {2046,
1000001, 121151}, {2047, 1000001, 121150}, {2048, 1000001, 121055}, {2049,
1000001, 121150}, {2050, 1000001, 121151}, {2051, 1000001, 121151}, {2052,
1000001, 121150}, {2053, 1000001, 121055}, {2054, 1000001, 121150}, {2055,
1000001, 121149}, {2056, 1000001, 121149}, {2057, 1000001, 121148}, {2058,
1000001, 121054}, {2059, 1000001, 121148}, {2060, 1000001, 121149}, {2061,
1000001, 121149}, {2062, 1000001, 121148}, {2063, 1000001, 121054}, {2064,
1000001, 121148}, {2065, 1000001, 121147}, {2066, 1000001, 121147}, {2067,
1000001, 121146}, {2068, 1000001, 121053}, {2069, 1000001, 121146}, {2070,
1000001, 121147}, {2071, 1000001, 121147}, {2072, 1000001, 121146}, {2073,
1000001, 121053}, {2074, 1000001, 121146}, {2075, 1000001, 121145}, {2076,
1000001, 121145}, {2077, 1000001, 121144}, {2078, 1000001, 121052}, {2079,
1000001, 121144}, {2080, 1000001, 121145}, {2081, 1000001, 121145}, {2082,
1000001, 121144}, {2083, 1000001, 121052}, {2084, 1000001, 121144}, {2085,
1000001, 121143}, {2086, 1000001, 121143}, {2087, 1000001, 121142}, {2088,
1000001, 121051}, {2089, 1000001, 121142}, ...
__________
...
{8000, 9991, 180}, {8001, 9991, 180}, {8002, 9989, 167}, {8003, 9989,
147}, {8004, 1000001, 120413}, {8005, 9991, 179}, {8006, 1000001, 120414}, {8007,
9989, 166}, {8008, 9991, 179}, {8009, 9989, 146}, {8010, 9991, 179}, {8011,
1000001, 120414}, {8012, 9989, 166}, {8013, 9991, 179}, {8014, 9989, 146},
{8015, 1000001, 120413}, {8016, 1000001, 120412}, {8017, 9989, 165}, {8018,
9991, 178}, {8019, 9991, 178}, {8020, 1000001, 120413}, {8021, 1000001,
120412}, {8022, 9989, 165}, {8023, 9991,
178}, {8024, 9991, 178}, {8025, 1000001, 120411}, {8026, 9989, 145}, {8027,
9989, 164}, {8028, 1000001, 120412}, {8029, 9991, 177}, {8030, 1000001,
120411}, {8031, 9989, 145}, {8032, 9989, 164}, {8033, 1000001, 120412}, {8034,
9991, 177}, {8035, 9989, 144}, {8036, 9991, 177}, {8037, 9989, 163}, {8038,
1000001, 120410}, {8039, 1000001, 120411}, {8040, 9989, 144}, {8041, 9991, 177}, {8042, 9989, 163},
{8043, 1000001, 120410}, {8044, 1000001, 120411}, {8045, 9991, 176}, {8046,
9991, 176}, {8047, 9989, 162}, {8048, 9989, 143}, {8049, 1000001, 120409},
{8050, 9991, 176}, {8051, 9991, 176}, {8052, 9989, 162}, {8053, 9989, 143},
{8054, 1000001, 120409}, {8055, 9991, 175}, {8056, 1000001, 120410}, {8057,
9989, 161}, {8058, 9991, 175}, {8059, 9989, 142}, {8060, 9991, 175}, {8061,
1000001, 120410}, {8062, 9989, 161}, {8063, 9991, 175}, {8064, 9989, 142},
{8065, 1000001, 120409}, {8066, 1000001, 120408}, {8067, 9989, 160}, {8068, 9991, 174}, {8069, 9991, 174},
{8070, 1000001, 120409}, {8071, 1000001, 120408}, {8072, 9989, 160}, {8073,
9991, 174}, {8074, 9991, 174}, {8075, 1000001, 120407}, {8076, 9989, 141},
{8077, 9989, 159}, {8078, 1000001, 120408}, {8079, 9991, 173}, {8080, 1000001,
120407}, {8081, 9989, 141}, {8082, 9989, 159}, {8083, 1000001, 120408}, {8084,
9991, 173}, {8085, 9989, 140}, {8086, 9991, 173}, {8087, 9989, 158}, {8088,
1000001, 120406}, {8089, 1000001, 120407}, {8090, 9989, 140}, {8091, 9991,
173}, {8092, 9989, 158}, {8093, 1000001, 120406}, {8094, 1000001, 120407},
{8095, 9991, 172}, {8096, 9991, 172}, {8097, 9989, 157}, {8098, 9989, 139},
{8099, 1000001, 120405}, {8100, 9991, 172}, {8101, 9991, 172}, {8102, 9989,
157}, {8103, 9989, 139}, {8104, 1000001, 120405}, {8105, 9991, 171}, {8106,
1000001, 120406}, {8107, 9989, 156}, {8108, 9991, 171}, {8109, 9989, 138},
{8110, 9991, 171}, {8111, 1000001, 120406}, {8112, 9989, 156}, {8113, 9991,
171}, {8114, 9989, 138}, {8115, 1000001, 120405}, {8116, 1000001, 120404}, {8117,
9989, 155}, {8118, 9991, 170}, {8119, 9991, 170}, {8120, 1000001, 120405},
{8121, 1000001, 120404}, {8122, 9989, 155}, {8123, 9991, 170}, {8124, 9991,
170}, {8125, 1000001, 120403}, {8126, 9989, 137}, {8127, 9989, 154}, {8128,
1000001, 120404}, {8129, 9991, 169}, {8130, 1000001, 120403}, {8131, 9989,
137}, {8132, 9989, 154}, {8133, 1000001, 120404}, {8134, 9991, 169}, {8135,
9989, 136}, {8136, 9991, 169}, {8137, 9989, 153}, {8138, 1000001, 120402},
{8139, 1000001, 120403}, {8140, 9989, 136}, {8141, 9991, 169}, {8142, 9989,
153}, {8143, 1000001, 120402}, {8144, 1000001, 120403}, {8145, 9991, 168}, {8146, 9991, 168}, {8147, 9989, 152},
{8148, 9989, 135}, {8149, 1000001, 120401}, {8150, 9991, 168}, {8151, 9991, 168},
{8152, 9989, 152}, {8153, 9989, 135}, {8154, 1000001, 120401}, {8155, 9991,
167}, {8156, 1000001, 120402}, {8157, 9989, 151}, {8158, 9991, 167}, {8159,
9989, 134}, {8160, 9991, 167}, {8161, 1000001, 120402}, {8162, 9989, 151},
{8163, 9991, 167}, {8164, 9989, 134}, {8165, 1000001, 120401}, {8166, 1000001,
120400}, {8167, 9989, 150}, {8168, 9991, 166}, {8169, 9991, 166}, {8170,
1000001, 120401}, {8171, 1000001, 120400}, {8172, 9989, 150}, {8173, 9991, 166}, {8174, 9991, 166},
{8175, 1000001, 120399}, {8176, 9989, 133}, {8177, 9989, 149}, {8178, 1000001,
120400}, {8179, 9991, 165}, {8180, 1000001, 120399}, {8181, 9989, 133}, {8182,
9989, 149}, {8183, 1000001, 120400}, {8184, 9991, 165}, {8185, 9989, 132},
{8186, 9991, 165}, {8187, 9989, 148}, {8188, 1000001, 120398}, {8189, 1000001,
120399}, {8190, 9989, 132}, {8191, 9991,
165}, {8192, 9989, 148}, {8193, 1000001, 120398}, {8194, 1000001, 120399},
{8195, 9991, 164}, {8196, 9991, 164}, {8197, 9989, 147}, {8198, 9989, 131},
{8199, 1000001, 120397}, {8200, 9991, 164}, {8201, 9991, 164}, {8202, 9989, 147},
{8203, 9989, 131}, {8204, 1000001, 120397}, {8205, 9991, 163}, {8206, 1000001,
120398}, {8207, 9989, 146}, {8208, 9991, 163}, {8209, 9989, 130}, {8210, 9991,
163}, {8211, 1000001, 120398}, {8212, 9989, 146}, {8213, 9991, 163}, {8214,
9989, 130}, {8215, 1000001, 120397}, {8216, 1000001, 120396}, {8217, 9989, 145}, {8218, 9991, 162}, {8219, 9991, 162},
{8220, 1000001, 120397}, {8221, 1000001, 120396}, {8222, 9989, 145}, {8223,
9991, 162}, {8224, 9991, 162}, {8225, 1000001, 120395}, {8226, 9989, 129}, {8227,
9989, 144}, {8228, 1000001, 120396}, {8229, 9991, 161}, {8230, 1000001,
120395}, {8231, 9989, 129}, {8232, 9989, 144}, {8233, 1000001, 120396}, {8234,
9991, 161}, {8235, 9989, 128}, {8236, 9991, 161}, {8237, 9989, 143}, {8238,
1000001, 120394}, {8239, 1000001, 120395}, {8240, 9989, 128}, {8241, 9991,
161}, {8242, 9989, 143}, {8243, 1000001, 120394}, {8244, 1000001, 120395},
{8245, 9991, 160}, {8246, 9991, 160}, {8247, 9989, 142}, {8248, 9989, 127},
{8249, 1000001, 120393}, {8250, 9991, 160}, {8251, 9991, 160}, {8252, 9989,
142}, {8253, 9989, 127}, {8254, 1000001, 120393}, {8255, 9991, 159}, {8256,
1000001, 120394}, {8257, 9989, 141}, {8258, 9991, 159}, {8259, 9989, 126},
{8260, 9991, 159}, {8261, 1000001, 120394}, {8262, 9989, 141}, {8263, 9991,
159}, {8264, 9989, 126}, {8265, 1000001, 120393}, {8266, 1000001, 120392},
{8267, 9989, 140}, {8268, 9991, 158}, {8269, 9991, 158}, {8270, 1000001,
120393}, {8271, 1000001, 120392}, {8272, 9989, 140}, {8273, 9991, 158}, {8274,
9991, 158}, {8275, 1000001, 120391}, {8276, 9989, 125}, {8277, 9989, 139},
{8278, 1000001, 120392}, {8279, 9991, 157}, {8280, 1000001, 120391}, {8281,
9989, 125}, {8282, 9989, 139}, {8283, 1000001, 120392}, {8284, 9991, 157},
{8285, 9989, 124}, {8286, 9991, 157}, {8287, 9989, 138}, {8288, 1000001,
120390}, {8289, 1000001, 120391}, {8290, 9989, 124}, {8291, 9991, 157}, {8292,
9989, 138}, {8293, 1000001, 120390}, {8294, 1000001, 120391}, {8295, 9991, 156}, {8296, 9991, 156}, {8297, 9989, 137},
{8298, 9989, 123}, {8299, 1000001, 120389}, {8300, 9991, 156}, {8301, 9991,
156}, {8302, 9989, 137}, {8303, 9989, 123}, {8304, 1000001, 120389}, {8305,
9991, 155}, {8306, 1000001, 120390}, {8307, 9989, 136}, {8308, 9991, 155},
{8309, 9989, 122}, {8310, 9991, 155}, {8311, 1000001, 120390}, {8312, 9989, 136},
{8313, 9991, 155}, {8314, 9989, 122}, {8315, 1000001, 120389}, {8316, 1000001,
120388}, {8317, 9989, 135}, {8318, 9991, 154}, {8319, 9991, 154}, {8320,
1000001, 120389}, {8321, 1000001, 120388}, {8322, 9989, 135}, {8323, 9991, 154}, {8324, 9991, 154},
{8325, 1000001, 120387}, {8326, 9989, 121}, {8327, 9989, 134}, {8328, 1000001,
120388}, {8329, 9991, 153}, {8330, 1000001, 120387}, {8331, 9989, 121}, {8332,
9989, 134}, {8333, 1000001, 120388}, {8334, 9991, 153}, {8335, 9989, 120},
{8336, 9991, 153}, {8337, 9989, 133}, {8338, 1000001, 120386}, {8339, 1000001,
120387}, {8340, 9989, 120}, {8341, 9991,
153}, {8342, 9989, 133}, {8343, 1000001, 120386}, {8344, 1000001, 120387},
{8345, 9991, 152}, {8346, 9991, 152}, {8347, 9989, 132}, {8348, 9989, 119},
{8349, 1000001, 120385}, {8350, 9991, 152}, {8351, 9991, 152}, {8352, 9989,
132}, {8353, 9989, 119}, {8354, 1000001, 120385}, {8355, 9991, 151}, {8356,
1000001, 120386}, {8357, 9989, 131}, {8358, 9991, 151}, {8359, 9989, 118},
{8360, 9991, 151}, {8361, 1000001, 120386}, {8362, 9989, 131}, {8363, 9991,
151}, {8364, 9989, 118}, {8365, 1000001, 120385}, {8366, 1000001, 120384},
{8367, 9989, 130}, {8368, 9991, 150},
{8369, 9991, 150}, {8370, 1000001, 120385}, {8371, 1000001, 120384}, {8372,
9989, 130}, {8373, 9991, 150}, {8374, 9991, 150}, {8375, 1000001, 120383},
{8376, 9989, 117}, {8377, 9989, 129}, {8378, 1000001, 120384}, {8379, 9991,
149}, {8380, 1000001, 120383}, {8381, 9989, 117}, {8382, 9989, 129}, {8383,
1000001, 120384}, {8384, 9991, 149}, {8385, 9989, 116}, {8386, 9991, 149},
{8387, 9989, 128}, {8388, 1000001, 120382}, {8389, 1000001, 120383}, {8390,
9989, 116}, {8391, 9991, 149}, {8392, 9989, 128}, {8393, 1000001, 120382},
{8394, 1000001, 120383}, {8395, 9991, 148}, {8396, 9991, 148}, {8397, 9989,
127}, {8398, 9989, 115}, {8399, 1000001, 120381}, {8400, 9991, 148}, {8401,
9991, 148}, {8402, 9989, 127}, {8403, 9989, 115}, {8404, 1000001, 120381},
{8405, 9991, 147}, {8406, 1000001, 120382}, {8407, 9989, 126}, {8408, 9991,
147}, {8409, 9989, 114}, {8410, 9991, 147}, {8411, 1000001, 120382}, {8412,
9989, 126}, {8413, 9991, 147}, {8414, 9989, 114}, {8415, 1000001, 120381},
{8416, 1000001, 120380}, {8417, 9989, 125}, {8418, 9991, 146}, {8419, 9991,
146}, {8420, 1000001, 120381}, {8421, 1000001, 120380}, {8422, 9989, 125},
{8423, 9991, 146}, {8424, 9991, 146}, {8425, 1000001, 120379}, {8426, 9989,
113}, {8427, 9989, 124}, {8428, 1000001, 120380}, {8429, 9991, 145}, {8430,
1000001, 120379}, {8431, 9989, 113}, {8432, 9989, 124}, {8433, 1000001,
120380}, {8434, 9991, 145}, {8435, 9989, 112}, {8436, 9991, 145}, {8437, 9989,
123}, {8438, 1000001, 120378}, {8439, 1000001, 120379}, {8440, 9989, 112},
{8441, 9991, 145}, {8442, 9989, 123}, {8443, 1000001, 120378}, {8444, 1000001,
120379}, {8445, 9991, 144}, {8446, 9991,
144}, {8447, 9989, 122}, {8448, 9989, 111}, {8449, 1000001, 120377}, {8450,
9991, 144}, {8451, 9991, 144}, {8452, 9989, 122}, {8453, 9989, 111}, {8454,
1000001, 120377}, {8455, 9991, 143}, {8456, 1000001, 120378}, {8457, 9989,
121}, {8458, 9991, 143}, {8459, 9989, 110}, {8460, 9991, 143}, {8461, 1000001,
120378}, {8462, 9989, 121}, {8463, 9991, 143}, {8464, 9989, 110}, {8465,
1000001, 120377}, {8466, 1000001, 120376}, {8467, 9989, 120}, {8468, 9991,
142}, {8469, 9991, 142}, {8470, 1000001, 120377}, {8471, 1000001, 120376},
{8472, 9989, 120}, {8473, 9991, 142},
{8474, 9991, 142}, {8475, 1000001, 120375}, {8476, 9989, 109}, {8477, 9989,
119}, {8478, 1000001, 120376}, {8479, 9991, 141}, {8480, 1000001, 120375},
{8481, 9989, 109}, {8482, 9989, 119}, {8483, 1000001, 120376}, {8484, 9991,
141}, {8485, 9989, 108}, {8486, 9991, 141}, {8487, 9989, 118}, {8488, 1000001,
120374}, {8489, 1000001, 120375}, {8490, 9989,
108}, {8491, 9991, 141}, {8492, 9989, 118}, {8493, 1000001, 120374},
{8494, 1000001, 120375}, {8495, 9991, 140}, {8496, 9991, 140}, {8497, 9989,
117}, {8498, 9989, 107}, {8499, 1000001, 120373}, {8500, 9991, 140}, {8501,
9991, 140}, {8502, 9989, 117}, {8503, 9989, 107}, {8504, 1000001, 120373},
{8505, 9991, 139}, {8506, 1000001, 120374}, {8507, 9989, 116}, {8508, 9991,
139}, {8509, 9989, 106}, {8510, 9991, 139}, {8511, 1000001, 120374}, {8512,
9989, 116}, {8513, 9991, 139}, {8514, 9989, 106}, {8515, 1000001, 120373},
{8516, 1000001, 120372}, {8517, 9989,
115}, {8518, 9991, 138}, {8519, 9991, 138}, {8520, 1000001, 120373},
{8521, 1000001, 120372}, {8522, 9989, 115}, {8523, 9991, 138}, {8524, 9991,
138}, {8525, 1000001, 120371}, {8526, 9989, 105}, {8527, 9989, 114}, {8528,
1000001, 120372}, {8529, 9991, 137}, {8530, 1000001, 120371}, {8531, 9989,
105}, {8532, 9989, 114}, {8533, 1000001, 120372}, {8534, 9991, 137}, {8535,
9989, 104}, {8536, 9991, 137}, {8537, 9989, 113}, {8538, 1000001, 120370},
{8539, 1000001, 120371}, {8540, 9989, 104}, {8541, 9991, 137}, {8542, 9989,
113}, {8543, 1000001, 120370}, {8544, 1000001, 120371}, {8545, 9991, 136},
{8546, 9991, 136}, {8547, 9989, 112}, {8548, 9989, 103}, {8549, 1000001,
120369}, {8550, 9991, 136}, {8551, 9991, 136}, {8552, 9989, 112}, {8553, 9989,
103}, {8554, 1000001, 120369}, {8555, 9991, 135}, {8556, 1000001, 120370},
{8557, 9989, 111}, {8558, 9991, 135}, {8559, 9989, 102}, {8560, 9991, 135},
{8561, 1000001, 120370}, {8562, 9989, 111}, {8563, 9991, 135}, {8564, 9989,
102}, {8565, 1000001, 120369}, {8566, 1000001, 120368}, {8567, 9989, 110},
{8568, 9991, 134}, {8569, 9991, 134}, {8570, 1000001, 120369}, {8571, 1000001,
120368}, {8572, 9989, 110}, {8573, 9991, 134}, {8574, 9991, 134}, {8575,
1000001, 120367}, {8576, 9989, 101}, {8577, 9989, 109}, {8578, 1000001,
120368}, {8579, 9991, 133}, {8580, 1000001, 120367}, {8581, 9989, 101}, {8582,
9989, 109}, {8583, 1000001, 120368}, {8584, 9991, 133}, {8585, 9989, 100},
{8586, 9991, 133}, {8587, 9989, 108}, {8588, 1000001, 120366}, {8589, 1000001,
120367}, {8590, 9989, 100}, {8591, 9991, 133}, {8592, 9989, 108}, {8593,
1000001, 120366}, {8594, 1000001, 120367}, {8595, 9991, 132}, {8596, 9991, 132}, {8597, 9989, 107},
{8598, 9989, 99}, {8599, 1000001, 120365}, {8600, 9991, 132}, {8601, 9991,
132}, {8602, 9989, 107}, {8603, 9989, 99}, {8604, 1000001, 120365}, {8605,
9991, 131}, {8606, 1000001, 120366}, {8607, 9989, 106}, {8608, 9991, 131},
{8609, 9989, 98}, {8610, 9991, 131}, {8611,
1000001, 120366}, {8612, 9989, 106}, {8613, 9991, 131}, {8614, 9989,
98}, {8615, 1000001, 120365}, {8616, 1000001, 120364}, {8617, 9989, 105},
{8618, 9991, 130}, {8619, 9991, 130}, {8620, 1000001, 120365}, {8621, 1000001,
120364}, {8622, 9989, 105}, {8623, 9991, 130}, {8624, 9991, 130}, {8625,
1000001, 120363}, {8626, 9989, 97}, {8627, 9989, 104}, {8628, 1000001, 120364},
{8629, 9991, 129}, {8630, 1000001, 120363}, {8631, 9989, 97}, {8632, 9989,
104}, {8633, 1000001, 120364}, {8634, 9991, 129}, {8635, 9989, 96}, {8636,
9991, 129}, {8637, 9989, 103}, {8638, 1000001, 120362}, {8639, 1000001, 120363}, {8640, 9989, 96}, {8641,
9991, 129}, {8642, 9989, 103}, {8643,
1000001, 120362}, {8644, 1000001, 120363}, {8645, 9991, 128}, {8646, 9991,
128}, {8647, 9989, 102}, {8648, 9989, 95}, {8649, 1000001, 120361}, {8650,
9991, 128}, {8651, 9991, 128}, {8652, 9989, 102}, {8653, 9989, 95}, {8654,
1000001, 120361}, {8655, 9991, 127}, {8656, 1000001, 120362}, {8657, 9989,
101}, {8658, 9991, 127}, {8659, 9989, 94}, {8660, 9991, 127}, {8661, 1000001, 120362}, {8662, 9989, 101}, {8663,
9991, 127}, {8664, 9989, 94}, {8665, 1000001, 120361}, {8666, 1000001, 120360},
{8667, 9989, 100}, {8668, 9991, 126}, {8669, 9991, 126}, {8670, 1000001,
120361}, {8671, 1000001, 120360}, {8672, 9989, 100}, {8673, 9991, 126}, {8674,
9991, 126}, {8675, 1000001, 120359}, {8676, 9989, 93}, {8677, 9989, 99}, {8678,
1000001, 120360}, {8679, 9991, 125}, {8680, 1000001, 120359}, {8681, 9989, 93},
{8682, 9989, 99}, {8683, 1000001, 120360}, {8684, 9991, 125}, {8685, 9989, 92},
{8686, 9991, 125}, {8687, 9989, 98}, {8688, 1000001, 120358}, {8689, 1000001,
120359}, {8690, 9989, 92}, {8691, 9991, 125}, {8692, 9989, 98}, {8693, 1000001,
120358}, {8694, 1000001, 120359}, {8695, 9991, 124}, {8696, 9991, 124}, {8697,
9989, 97}, {8698, 9989, 91}, {8699, 1000001, 120357}, {8700, 9991, 124}, {8701,
9991, 124}, {8702, 9989, 97}, {8703, 9989, 91}, {8704, 1000001, 120357}, {8705,
9991, 123}, {8706, 1000001, 120358}, {8707,
9989, 96}, {8708, 9991, 123}, {8709, 9989, 90}, {8710, 9991, 123},
{8711, 1000001, 120358}, {8712, 9989, 96}, {8713, 9991, 123}, {8714, 9989, 90},
{8715, 1000001, 120357}, {8716, 1000001, 120356}, {8717, 9989, 95}, {8718,
9991, 122}, {8719, 9991, 122}, {8720, 1000001, 120357}, {8721, 1000001,
120356}, {8722, 9989, 95}, {8723, 9991,
122}, {8724, 9991, 122}, {8725, 1000001, 120355}, {8726, 9989, 89}, {8727,
9989, 94}, {8728, 1000001, 120356}, {8729, 9991, 121}, {8730, 1000001, 120355},
{8731, 9989, 89}, {8732, 9989, 94}, {8733, 1000001, 120356}, {8734, 9991, 121},
{8735, 9989, 88}, {8736, 9991, 121}, {8737, 9989, 93}, {8738, 1000001, 120354},
{8739, 1000001, 120355}, {8740, 9989, 88}, {8741, 9991, 121}, {8742, 9989, 93},
{8743, 1000001, 120354}, {8744, 1000001, 120355}, {8745, 9991, 120}, {8746,
9991, 120}, {8747, 9989, 92}, {8748,
9989, 87}, {8749, 1000001, 120353}, {8750, 9991, 120}, {8751, 9991, 120},
{8752, 9989, 92}, {8753, 9989, 87}, {8754, 1000001, 120353}, {8755, 9991, 119},
{8756, 1000001, 120354}, {8757, 9989,
91}, {8758, 9991, 119}, {8759, 9989, 86}, {8760, 9991, 119}, {8761, 1000001,
120354}, {8762, 9989, 91}, {8763, 9991, 119}, {8764, 9989, 86}, {8765, 1000001,
120353}, {8766, 1000001, 120352}, {8767, 9989, 90}, {8768, 9991, 118}, {8769,
9991, 118}, {8770, 1000001, 120353}, {8771, 1000001, 120352}, {8772, 9989, 90}, {8773, 9991, 118}, {8774, 9991, 118},
{8775, 1000001, 120351}, {8776, 9989, 85}, {8777, 9989, 89}, {8778, 1000001,
120352}, {8779, 9991, 117}, {8780, 1000001, 120351}, {8781, 9989, 85}, {8782,
9989, 89}, {8783, 1000001, 120352}, {8784, 9991, 117}, {8785, 9989, 84}, {8786,
9991, 117}, {8787, 9989, 88}, {8788, 1000001, 120350}, {8789, 1000001, 120351},
{8790, 9989, 84}, {8791, 9991, 117}, {8792, 9989, 88}, {8793, 1000001, 120350},
{8794, 1000001, 120351}, {8795, 9991, 116}, {8796, 9991, 116}, {8797,
9989, 87}, {8798, 9989, 83}, {8799,
1000001, 120349}, {8800, 9991, 116}, {8801, 9991, 116}, {8802, 9989, 87},
{8803, 9989, 83}, {8804, 1000001, 120349}, {8805, 9991, 115}, {8806, 1000001,
120350}, {8807, 9989, 86}, {8808, 9991,
115}, {8809, 9989, 82}, {8810, 9991, 115}, {8811, 1000001, 120350}, {8812,
9989, 86}, {8813, 9991, 115}, {8814, 9989, 82}, {8815, 1000001, 120349}, {8816,
1000001, 120348}, {8817, 9989, 85}, {8818, 9991, 114}, {8819, 9991, 114},
{8820, 1000001, 120349}, {8821, 1000001, 120348}, {8822, 9989, 85}, {8823, 9991, 114}, {8824, 9991, 114},
{8825, 1000001, 120347}, {8826, 9989, 81}, {8827, 9989, 84}, {8828, 1000001,
120348}, {8829, 9991, 113}, {8830, 1000001, 120347}, {8831, 9989, 81}, {8832,
9989, 84}, {8833, 1000001, 120348}, {8834, 9991, 113}, {8835, 9989, 80}, {8836,
9991, 113}, {8837, 9989, 83}, {8838, 1000001, 120346}, {8839, 1000001, 120347},
{8840, 9989, 80}, {8841, 9991, 113}, {8842, 9989, 83}, {8843, 1000001, 120346},
{8844, 1000001, 120347}, {8845, 9991, 112}, {8846, 9991, 112}, {8847,
9989, 82}, {8848, 9989, 79}, {8849,
1000001, 120345}, {8850, 9991, 112}, {8851, 9991, 112}, {8852, 9989, 82},
{8853, 9989, 79}, {8854, 1000001, 120345}, {8855, 9991, 111}, {8856, 1000001,
120346}, {8857, 9989, 81}, {8858, 9991,
111}, {8859, 9989, 78}, {8860, 9991, 111}, {8861, 1000001, 120346}, {8862,
9989, 81}, {8863, 9991, 111}, {8864, 9989, 78}, {8865, 1000001, 120345}, {8866,
1000001, 120344}, {8867, 9989, 80}, {8868, 9991, 110}, {8869, 9991, 110},
{8870, 1000001, 120345}, {8871, 1000001, 120344}, {8872, 9989, 80}, {8873, 9991, 110}, {8874, 9991, 110},
{8875, 1000001, 120343}, {8876, 9989, 77}, {8877, 9989, 79}, {8878, 1000001,
120344}, {8879, 9991, 109}, {8880, 1000001, 120343}, {8881, 9989, 77}, {8882,
9989, 79}, {8883, 1000001, 120344}, {8884, 9991, 109}, {8885, 9989, 76}, {8886,
9991, 109}, {8887, 9989, 78}, {8888, 1000001, 120342}, {8889, 1000001, 120343},
{8890, 9989, 76}, {8891, 9991, 109}, {8892, 9989, 78}, {8893, 1000001, 120342},
{8894, 1000001, 120343}, {8895, 9991, 108}, {8896, 9991, 108}, {8897,
9989, 77}, {8898, 9989, 75}, {8899,
1000001, 120341}, {8900, 9991, 108}, {8901, 9991, 108}, {8902, 9989, 77},
{8903, 9989, 75}, {8904, 1000001, 120341}, {8905, 9991, 107}, {8906, 1000001,
120342}, {8907, 9989, 76}, {8908, 9991,
107}, {8909, 9989, 74}, {8910, 9991, 107}, {8911, 1000001, 120342}, {8912,
9989, 76}, {8913, 9991, 107}, {8914, 9989, 74}, {8915, 1000001, 120341}, {8916,
1000001, 120340}, {8917, 9989, 75}, {8918, 9991, 106}, {8919, 9991, 106},
{8920, 1000001, 120341}, {8921, 1000001, 120340}, {8922, 9989, 75}, {8923, 9991, 106}, {8924, 9991, 106},
{8925, 1000001, 120339}, {8926, 9989, 73}, {8927, 9989, 74}, {8928, 1000001,
120340}, {8929, 9991, 105}, {8930, 1000001, 120339}, {8931, 9989, 73}, {8932,
9989, 74}, {8933, 1000001, 120340}, {8934, 9991, 105}, {8935, 9989, 72}, {8936,
9991, 105}, {8937, 9989, 73}, {8938, 1000001, 120338}, {8939, 1000001, 120339},
{8940, 9989, 72}, {8941, 9991, 105}, {8942, 9989, 73}, {8943, 1000001, 120338},
{8944, 1000001, 120339}, {8945, 9991, 104}, {8946, 9991, 104}, {8947,
9989, 72}, {8948, 9989, 71}, {8949,
1000001, 120337}, {8950, 9991, 104}, {8951, 9991, 104}, {8952, 9989, 72}, {8953,
9989, 71}, {8954, 1000001, 120337}, {8955, 9991, 103}, {8956, 1000001, 120338},
{8957, 9989, 71}, {8958, 9991, 103},
{8959, 9989, 70}, {8960, 9991, 103}, {8961, 1000001, 120338}, {8962, 9989, 71},
{8963, 9991, 103}, {8964, 9989, 70}, {8965, 1000001, 120337}, {8966, 1000001,
120336}, {8967, 9989, 70}, {8968, 9991, 102}, {8969, 9991, 102}, {8970,
1000001, 120337}, {8971, 1000001, 120336}, {8972, 9989, 70}, {8973, 9991, 102}, {8974, 9991, 102},
{8975, 1000001, 120335}, {8976, 9989, 69}, {8977, 9989, 69}, {8978, 1000001,
120336}, {8979, 9991, 101}, {8980, 1000001, 120335}, {8981, 9989, 69}, {8982,
9989, 69}, {8983, 1000001, 120336}, {8984, 9991, 101}, {8985, 9989, 68}, {8986,
9991, 101}, {8987, 9989, 68}, {8988, 1000001, 120334}, {8989, 1000001, 120335},
{8990, 9989, 68}, {8991, 9991, 101}, {8992, 9989, 68}, {8993, 1000001, 120334},
{8994, 1000001, 120335}, {8995, 9987, 67}, {8996, 9991, 100}, {8997, 1000001,
119334}, {8998, 9989, 67}, {8999, 9993, 67}, {9000, 9987, 67}, {9001, 9991,
100}, {9002, 1000001, 119334}, {9003, 9989, 67}, {9004, 9993, 67}, {9005,
1000001, 120333}, {9006, 9991, 99}, {9007, 1000001, 120334}, {9008, 1000001,
119333}, {9009, 9987, 66}, {9010, 1000001, 120333}, {9011, 9991, 99}, {9012,
1000001, 120334}, {9013, 1000001, 119333}, {9014, 9987, 66}, {9015, 9989, 66},
{9016, 9991, 98}, {9017, 9993, 66}, {9018, 1000001, 120333}}
Aai – a complement:
Maybe the following is of some interest too.
If I’m not wrong the terminator values are (well-) known through the
following patterns:
86 87 88 89 91 92 93 94
986 987 988 989 991 992 993 994
9986 9987 9988 9989 9991 9992 9993 9994
etc.
The next row is made by prepending a 9 to
numbers of the previous row.
By working with predecessors we are able to build a complete tree of
each terminator value. Each branch of such a tree is then one of Eric’s S sequences, e.g. take
terminator value 987:
┌───────────────────────────┐
│987 974 │
├───────────────────────────┤
│987 969 960 │
├───────────────────────────┤
│987 969 955 943 932 │
├───────────────────────────┤
│987 969 955 943 927 914 │
├───────────────────────────┤
│987 969 955 943 927 909 900│
├───────────────────────────┤
│987 969 955 943 927 909 895│
├───────────────────────────┤
│987 969 955 938 │
└───────────────────────────┘
Count of ‘all’ branches of terminator 989: 330
Some end-leaves are counted more than once, e.g. 14 (has
2 successors):
19 31 ___
14
/ \ 35 44 53 62 71 79 97 105 111 113
117 125 131 --> 989
\ 20 22 26 /
But 14 has a smaller terminator than 989,
because:
19 31 ___ / 80 88
14 / \ 35 44 53 62 71
\ 20 22 26 / \ 79 97 105 111 113 117 125 131
--> 989
Unique starting values that end with 989 are:
5 6 7 8 9 10 12 14 16 18 21 23 25 27 29 30 32 34
36 38 41 43 45 47 50 52 56 58 61 63 78 83 85 90 96 98
100 102 104 106 108 110 112 114 116 118 120 122 124 126 128 130 132 134
136 138 140 142 144 146 148 150 152 154 156 158 160 162 164 166 168 170
172 174 176 178 180 182 184 186 188 190 192 194 196 198 201 203 205 207
209 211 213 215 217 219 221 223 225 227 229 231 233 235 237 239 241 243
245 247 249 251 253 255 257 259 261 263 265 267 269 271 273 275 277 279
281 283 285 287 289 291 293 295 297 299 300 302 304 306 308 310 312 314
316 318 320 322 324 326 328 330 332 334 336 338 340 342 344 346 348 350
352 354 356 358 360 362 364 366 368 370 372 374 376 378 380 382 384 386
388 390 392 394 407 413 415 419 423 425 427 429 437 443 445 449 453 455
457 459 467 473 475 479 483 485 487 489 499 504 506 508 516 522 532 534
538 544 546 548 556 562 572 574 578 584 586 588 598 603 605 607 615 621
631 633 637 643 645 647 655 661 671 673 677 683 685 687 697 702 706 714
724 730 740 742 746 752 756 764 774 780 790 792 798 803 807 809 817 827
831 835 837 847 853 857 859 867 877 881 885 887 920 934 952 958 980
Example 504
504 513 521 527 539 553 561
567 579 593 602 610 616 628
642 650 656 668 682 690 696
709 725 737 751 759 775 787
802 812 822 832 842 852 862
872 882 892 903 915 929 947
963 975 989
Be aware: all of this is based on the assumption that there are only
certain terminator values. :-)
--
Met vriendelijke groet,
=@@i
Claudio Meller:
Hi Eric, I found those numbers with no predecessors :
1, 2, 3, 4, 5, 6, 7, 8, 9,
10, 12, 14, 16, 18, 21, 23, 25, 27, 29, 30, 32, 34, 36, 38, 41, 43,
45, 47, 49, 50, 52, 54, 56, 58, 61, 63, 65, 67, 69, 70, 72, 74, 76,
78, 81, 83, 85, 87, 89, 90, 92, 94, 96, 98, 100, 102, 104, 106, 108,
110, 112, 114, 116, 118, 120, 122, 124, 126, 128, 130, 132, 134, 136,
138, 140, 142, 144, 146, 148, 150, 152, 154, 156, 158, 160, 162, 164,
166, 168, 170, 172, 174, 176, 178, 180, 182, 184, 186, 188, 190, 192,
194, 196, 198, 201, 203, 205, 207, 209, 211, 213, 215, 217, 219, 221,
223, 225, 227, 229, 231, 233, 235, 237, 239, 241, 243, 245, 247, 249,
251, 253, 255, 257, 259, 261, 263, 265, 267, 269, 271, 273, 275, 277,
279, 281, 283, 285, 287, 289, 291, 293, 295, 297, 299, 300, 302, 304,
306, 308, 310, 312, 314, 316, 318, 320, 322, 324, 326, 328, 330, 332,
334, 336, 338, 340, 342, 344, 346, 348, 350, 352, 354, 356, 358, 360,
362, 364, 366, 368, 370, 372, 374, 376, 378, 380, 382, 384, 386, 388,
390, 392, 394, 396, 398, 401, 403,
405, 407, 409, 411, 413, 415, 417,
419, 421, 423, 425, 427, 429, 431, 433, 435, 437, 439, 441, 443, 445,
447, 449, 451, 453, 455, 457, 459, 461, 463, 465, 467, 469, 471, 473,
475, 477, 479, 481, 483, 485, 487, 489, 491, 493, 495, 497, 499, 500,
502, 504, 506, 508, 510, 512, 514, 516, 518, 520, 522, 524, 526, 528,
530, 532, 534, 536, 538, 540, 542, 544, 546, 548, 550, 552, 554, 556,
558, 560, 562, 564, 566, 568, 570, 572, 574, 576, 578, 580, 582, 584,
586, 588, 590, 592, 594, 596, 598, 601, 603, 605, 607, 609, 611, 613,
615, 617, 619, 621, 623, 625, 627, 629, 631, 633, 635, 637, 639, 641,
643, 645, 647, 649, 651, 653, 655, 657, 659, 661, 663, 665, 667, 669,
671, 673, 675, 677, 679, 681, 683, 685, 687, 689, 691, 693, 695, 697,
699, 700, 702, 704, 706, 708, 710, 712, 714, 716, 718, 720, 722, 724,
726, 728, 730, 732, 734, 736, 738, 740, 742, 744, 746, 748, 750, 752,
754, 756, 758, 760, 762, 764, 766, 768, 770, 772, 774, 776, 778, 780,
782, 784, 786, 788, 790, 792, 794, 796, 798, 801, 803, 805, 807, 809,
811, 813, 815, 817, 819, 821, 823, 825, 827, 829, 831, 833, 835, 837,
839, 841, 843, 845, 847, 849, 851, 853, 855, 857, 859, 861, 863, 865,
867, 869, 871, 873, 875, 877, 879, 881, 883, 885, 887, 889, 891, 893,
895, 897, 899, 900, 902, 904, 906, 908, 910, 912, 914, 916, 918, 920,
922, 924, 926, 928, 930, 932, 934, 936, 938, 940, 942, 944, 946, 948,
950, 952, 954, 956, 958, 960, 962, 964, 966, 968, 970, 972, 974, 976,
978, 980, 982, 984, 986, 988, 990, 992, 994, 996, 998, 1000, ...
... an integer beginning with
an odd digit has no predecessor if it ends also with an even digit (and vice
versa).
Aai se propose (en néerlandais et mathématiques), de démontrer que :
> (...) ca. half of the numbers of [1
.. N] have no predecessors recognized by the property
that the sum of leading and trailing digit has to be odd (a bit restated but
also mentioned by Claudio Meller). I hope that a mathematician will have a critical look
at the following :-)
Hoofdstelling:
Van de verzameling getallen
1 t/m N hebben (1 + N - 9)/2 getallen
geen voorganger.
Hulpstelling:
Elk getal
waarvan de som van het eerste en het laatste cijfer oneven is heeft geen voorganger
n-1
Gegeven: G = Sigma a(i)*10^i en a(n-1) + a(0) = 2*k + 1
0
Bewijs:
n-1
Stel voorganger P = Sigma b(i)*10^i met P < G, dan
is:
0
Bedenk dat van P
b(n-1) = 0 kan zijn.
n-1
G - P = a(n-1) + b(0) = Sigma
(a(i)-b(i))*10^i + a(0) - b(0)
1
n-1
Sigma (a(i)-b(i))*10^i - 2*b(0) = a(n-1) - a(0)
1
of druk beide zijden
uit in even/oneven expressies
2*r = 2*s + 1 is niet waar
--> P is geen voorganger
van G
q.e.d
Hieruit volgt tevens de hoofdstelling, ervan uitgaande dat 0 als opvolgers
1 t/m
9 heeft.
--
Met vriendelijke groet,
=@@i
Aai (later):
A consequence of the fact, that numbers with an odd sum of msd and lsd have no predecessors,
is:
- only half of the ‘terminator’-numbers are
real terminators. A number like 986 is an isolated number, no successors as
well as no predecessors.
Aai today (October 19th, 2011, upon a search for “solitaire”
numbers – integers with no predecessor and no successor):
In fact it’s not difficult to produce them (pen & paper). Look at
the following matrix of (potential) root (or terminator) numbers:
86 87
88 89 91
92 93 94
986 987
988 989 991
992 993 994
9986 9987 9988 9989 9991 9992
9993 9994
From each row, 4 of them are root numbers and the other 4 are
solitaires. Because the pattern of these numbers is obvious, extending the
matrix is easy.
E.g. the 5-th row will be (adding 9’s at the beginning):
999986 999987 999988 999989 999991 999992 999993 999994
From these rows only numbers with an odd sum of msd
and lsd are “solitaires”.
Here are the first few (are they really of some interest?)
87 89 92 94
986 988 992 994
9986 9988 9992 9994
99986 99988 99992 99994
999986 999988 999992 999994
9999986 9999988 9999992 9999994
99999986 99999988 99999992 99999994
999999986 999999988 999999992
999999994
9999999986 9999999988 9999999992
9999999994
99999999986 99999999988 99999999992
99999999994
999999999986 999999999988 999999999992
999999999994
...
I checked the missing numbers of the graph
in the interval [0...
1000] and they are:
87 89 92 94 986 988 992 994
... as expected.
_________
[FRENCH] :
On a compris le procédé :
- deux nombres séparés par une
virgule diffèrent d’une quantité égale à la somme des chiffres qui touchent la
virgule;
- la plupart des « petits nombres »
produisent des suites finies qui bloquent sur 989 (ou d’autres);
- 396
semble être le plus petit entier ne bloquant pas ;
- certain entiers n’ont pas de
prédécesseur, d’autres pas de successeur, d’autres ni l’un ni l’autre (le
dernier tableau ci-dessus, celui des « solitaires » proposé par Aai)
__________
A remark from Aai about his tree:
Also, it’s probably obvious, but the picture
suggests that numbers that end in e.g. 1001 are also trees. That’s of course
not true. The graph is about the interval [0 ... 999] and numbers > 999 are
kind of picture overshoot and do not tell how they continue.
(...)
And this last message, also by Aai:
I think I have a more or less clear picture of
this coupling of numbers:
I conjecture that all commasum
sequences are or finite because they end in a root number (has no successors,
e.g. 989) or are infinite because they start as branches (with one or more
members) of an infinite tree of which the smallest source number (starts a commasum sequence) is equal to 396.
Per interval of 10^n ..
10^(n+1) - 1 you can distinguish 4 finite trees and 4 solitaires,
all the other numbers are part of an infinite tree.
I propose that if a sequence has the choice of
joining the ‘infinite’ seq. or a seq. that ends in a root, then the seq. is
finite.
I checked the numbers of the graph that run out
of the picture, e.g. starting with 996 --> 1003: they all join the sequence
that starts with 396 at some point in the next interval.
But I still cannot proof that 396 runs forever.
--
Met vriendelijke groet,
=@@i
__________
Thank you to
Jack, Lars, Aai, Mark, Jean-Marc, Nicolas
Graner and Claudio for their
comments & computations.
__________