Copies to the right
Hello SeqFans,
Let’s
imagine this procedure for an integer “n” to reproduce itself.
Take “n” =
1234 for instance.
Step 1:
Print 1234
on a squared paper with one digit per cell. By convention we say that the first
digit of “n” occupies the cell of rank 1, the second digit of “n” the cell of
rank 2, etc.
Example of
step 1:
1 2 3 4 . . . . . .
. . . . . . . . . . . . . . . . . . . .
Step 2:
Create an
infinite amount of copies of each digit “d” of “n” to the right, with those
simple rules:
a) if the
digit “d” occupies the cell with rank “c”, then the next copies of “d” will
successively appear in the cells of rank “c+(d+1)”, “c+2(d+1)”, “c+3(d+1)”,
“c+4(d+1)”, ... “c+k(d+1)”.
b) if a cell is occupied by more than one digit, keep only the
biggest one.
Example of
step 2 (beginning of the infinite duplication):
1 2 3 4 2 . 3 2 4 . 3 .
1 4 3 . 2 . 4 2 1 . 3 4 1 2 3 . 4
. 3 2 1 4 3 . 1 2 4 . 2 . 3 4 1 . (etc.)
Some squares
won’t be filled, but this is not important. The important question is: when
does the initial “n” show again? This “when” will be measured by a “distance”
-- which will simply be the amount of cells the first digit of “n” has to “pass
through” to reach the cell occupied by its first “sound” copy.
If I’m not
wrong, the copy of 1234 is 60 cells away from its original:
1 2 3 4 2 . 3 2 4 . 3 .
1 4 3 . 2 . 4 2 1 . 3 4 1 2 3 . 4
. 3 2 1 4 3 . 1 2 4 . 2 . 3 4 1 . 3 .
4 2 1 . 2 4 3 2 1 . 4 . 1 2
3 4 (hit)
Two last
remarks:
Some
integers “vanish” immediately, they cannot reproduce; see for instance what
happens to 100:
Step 1:
1 0 0
Step 2:
1 0 1 0 1 0
1 0 1 0 1 0 1 0 1 0 1 0 ...
“100” has
vanished in step 2; there is no “distance” at all here (we could say by convention
that its distance is equal to zero).
Some
integers don’t vanish – but they don’t copy either, as 1011 and 1012, for instance (thanks to Hans Havermann); their
distance is set to zero too:
Step 1:
1 0 1 2
Step 2:
1 0 1 2 1 0 2 0 1 2
1 0 2 0 1 2 1 0 2 0 1 2 1 0 2 0 1 2 1 0 2 ...
__________
Ok, the tool
is ready. Let’s see a few [distances] for “n” to reproduce:
“n” [distance]
0 0 [1]
1 . 1 [2]
2 . . 2 [3]
3
. . . 3 [4]
4
. . . . 4 [5]
5
. . . . . 5 [3]
6
. . . . . . 6 [7]
7
. . . . . . . 7 [8]
8
. . . . . . . . 8 [9]
9
. . . . . . . . . 9 [10]
1 0 1 0 [2]
1 1 1 [1]
1 2 1
. 2 . 1 2
[6]
1 3 1
. 1 3 [4]
1 4 1
. 1 . 4 . 1 . 1 4
[10]
1 5 1
. 1 . 1 5
[6]
1 6 1
. 1 . 1 . 6 . 1 . 1 .
1 6 [14]
1 7 1
. 1 . 1 . 1 7 [8]
1 8 1
. 1 . 1 . 1 . 8 . 1 .
1 . 1 . 1 8 [18]
1 9 1
. 1 . 1 . 1 . 1 9
[10]
2 0 0 2 0
[3]
2 1 [6] (the integers “ab” and “ba”
share the same distance)
2 2 . 2 2 [3]
2 3
. 2 . 3 2 . . 3 . . 2 3 [12]
2 4
. 2 . . 4 . . 2 . 4 2 .
. 2 4 [15]
2 5
. 2 . . 2 5 [6]
2 6
. 2 . . 2 . 6 2 . . 2 .
. 6 . . 2 . . 2 6 [21]
2 7
. 2 . . 2 . . 7 . . 2 .
. 2 . 7 2 . . 2 . . 2 7
[24]
2 8
. 2 . . 2 . . 2 8
[9]
2 9
. 2 . . 2 . . 2 . 9 2 .
. 2 . . 2 . . 9 . . 2 . . 2 .
. 2 9 [30]
3 0 0 0 3 0 [4]
3 1 [4]
3 2 [12]
3 3 . . 3 3 [4]
3 4
. . 3 . 4 . 3 . . 4 3 . . . 4 . . . 3 4 [20]
3 5
. . 3 . . 5 3 .
. . 3 5 [12]
3 6
. . 3 . . . 6 . . . 3 .
. 6 3 . . . 3 . 6 . 3 . . . 3 6 [28]
3 7
. . 3 . . . 3 7 [8]
3 8
. . 3 . . . 3 . 8 . 3 . . . 3 . . 8 3 . . . 3 . . .
8 . . . 3 . . . 3 8 [36]
3 9
. . 3 . . . 3 .
. 9 3 . . . 3 . . . 3 9 [20]
4 0 0 0 0 4 0 [5]
4 1 [10]
4 2 [15]
4 3 [20]
...
Sequences
associated with this idea could be:
- Integers
which are their own distance (like 18)
- Distance(n)
- Smallest
integer with distance “n”
- Integers
with zero distance...
Best,
É.
__________
Hans
Havermann was quick to answer...
[Hans Havermann]:
> - Distance(n) :
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 2, 1, 6, 4, 10, 6, 14, 8, 18, 10, 3, 6, 3, 12,
15, 6, 21, 24, 9, 30, 4, 4, 12, 4, 20, 12, 28, 8, 36, 20, 5, 10, 15, 20, 5, 30,
35, 40, 45, 10, 6, 6, 6, 12, 30, 6, 42, 24, 18, 30, 7, 14, 21, 28, 35, 42, 7,
56, 63, 70, 8, 8, 24, 8, 40, 24, 56, 8, 72, 40, 9, 18, 9, 36, 45, 18, 63, 72,
9, 90, 10, 10, 30, 20, 10, 30, 70, 40, 90, 10, 0, 2, 6, 4, 10, 6, 14, 8, 18,
10, 0, 1, 3, 4, 5, 6, 7, 8, 9, 10, 0, 6, 6, 12, 30, 6, 42, 24, 18, 30, 0, 4,
12, 4, 20, 12, 28, 8, 36, 20, 0, 10, 30, 20, 10, 30, 70, 40, 90, 10, 0, 6, 6,
12, 30, 6, 42, 24, 18, 30, 0, 14, 42, 28, 70, 42, 14, 56, 126, 70, 0, 8, 24, 8,
40, 24, 56, 8, 72, 40, 0, 18, 18, 36, 90, 18, 126, 72, 18, 90, 0, 10, 30, 20,
10, 30, 70, 40, 90, 10, 3, 6, 3, 12, 15, 6, 21, 24, 9, 30, 6, 3, 6, 12, 30, 6,
42, 24, 18, 30, 3, 6, 1, 12, 15, 6, 21, 24, 9, 30, 12, 12, 12, 12, 60, 12, 84,
24, 36, 60, 15, 30, 15, 60, 15, 30, 105, 120, 45, 30, 6, 6, 6, 12, 30, 6, 42,
24, 18, 30, 21, 42, 21, 84, 105, 42, 21, 168, 63, 210, 24, 24, 24, 24, 120, 24,
168, 24, 72, 120, 9, 18, 9, 36, 45, 18, 63, 72, 9, 90, 30, 30, 30, 60, 30, 30,
210, 120, 90, 30, 4, 4, 12, 2, 20, 12, 28, 8, 36, 20, 4, 4, 12, 2, 20, 12, 28,
8, 36, 20, 12, 12, 12, 6, 60, 12, 84, 24, 36, 60, 4, 4, 12, 4, 20, 12, 28, 8,
36, 20, 20, 20, 60, 10, 20, 60, 140, 40, 180, 20, 12, 12, 12, 6, 60, 12, 84,
24, 36, 60, 28, 28, 84, 14, 140, 84, 28, 56, 252, 140, 8, 8, 24, 8, 40, 24, 56,
8, 72, 40, 36, 36, 36, 18, 180, 36, 252, 72, 36, 180, 20, 20, 60, 10, 20, 60,
140, 40, 180, 20, 5, 10, 15, 20, 5, 30, 35, 40, 45, 10, 10, 5, 30, 20, 10, 30,
70, 40, 90, 10, 15, 30, 15, 60, 15, 30, 105, 120, 45, 30, 20, 20, 60, 20, 20,
60, 140, 40, 180, 20, 5, 10, 15, 20, 5, 30, 35, 40, 45, 10, 30, 30, 30, 60, 30,
30, 210, 120, 90, 30, 35, 70, 105, 140, 35, 210, 35, 280, 315, 70, 40, 40, 120,
40, 40, 120, 280, 40, 360, 40, 45, 90, 45, 180, 45, 90, 315, 360, 45, 90, 10,
10, 30, 20, 10, 30, 70, 40, 90, 10, 6, 6, 6, 12, 30, 6, 42, 24, 18, 30, 6, 6,
6, 12, 30, 6, 42, 24, 18, 30, 6, 6, 6, 12, 30, 6, 42, 24, 18, 30, 12, 12, 12,
12, 60, 12, 84, 24, 36, 60, 30, 30, 30, 60, 30, 30, 210, 120, 90, 30, 6, 6, 6,
12, 30, 6, 42, 24, 18, 30, 42, 42, 42, 84, 210, 42, 42, 168, 126, 210, 24, 24,
24, 24, 120, 24, 168, 24, 72, 120, 18, 18, 18, 36, 90, 18, 126, 72, 18, 90, 30,
30, 30, 60, 30, 30, 210, 120, 90, 30, 7, 14, 21, 28, 35, 42, 7, 56, 63, 70, 14,
7, 42, 28, 70, 42, 14, 56, 126, 70, 21, 42, 21, 84, 105, 42, 21, 168, 63, 210,
28, 28, 84, 28, 140, 84, 28, 56, 252, 140, 35, 70, 105, 140, 35, 210, 35, 280,
315, 70, 42, 42, 42, 84, 210, 42, 42, 168, 126, 210, 7, 14, 21, 28, 35, 42, 7,
56, 63, 70, 56, 56, 168, 56, 280, 168, 56, 56, 504, 280, 63, 126, 63, 252, 315,
126, 63, 504, 63, 630, 70, 70, 210, 140, 70, 210, 70, 280, 630, 70, 8, 8, 24,
8, 40, 24, 56, 8, 72, 40, 8, 8, 24, 8, 40, 24, 56, 8, 72, 40, 24, 24, 24, 24,
120, 24, 168, 24, 72, 120, 8, 8, 24, 8, 40, 24, 56, 8, 72, 40, 40, 40, 120, 40,
40, 120, 280, 40, 360, 40, 24, 24, 24, 24, 120, 24, 168, 24, 72, 120, 56, 56,
168, 56, 280, 168, 56, 56, 504, 280, 8, 8, 24, 8, 40, 24, 56, 8, 72, 40, 72,
72, 72, 72, 360, 72, 504, 72, 72, 360, 40, 40, 120, 40, 40, 120, 280, 40, 360,
40, 9, 18, 9, 36, 45, 18, 63, 72, 9, 90, 18, 9, 18, 36, 90, 18, 126, 72, 18,
90, 9, 18, 9, 36, 45, 18, 63, 72, 9, 90, 36, 36, 36, 36, 180, 36, 252, 72, 36,
180, 45, 90, 45, 180, 45, 90, 315, 360, 45, 90, 18, 18, 18, 36, 90, 18, 126,
72, 18, 90, 63, 126, 63, 252, 315, 126, 63, 504, 63, 630, 72, 72, 72, 72, 360,
72, 504, 72, 72, 360, 9, 18, 9, 36, 45, 18, 63, 72, 9, 90, 90, 90, 90, 180, 90,
90, 630, 360, 90, 90, 10, 10, 30, 20, 10, 30, 70, 40, 90, 10, 10, 10, 30, 20,
10, 30, 70, 40, 90, 10, 30, 30, 30, 60, 30, 30, 210, 120, 90, 30, 20, 20, 60,
20, 20, 60, 140, 40, 180, 20, 10, 10, 30, 20, 10, 30, 70, 40, 90, 10, 30, 30,
30, 60, 30, 30, 210, 120, 90, 30, 70, 70, 210, 140, 70, 210, 70, 280, 630, 70,
40, 40, 120, 40, 40, 120, 280, 40, 360, 40, 90, 90, 90, 180, 90, 90, 630, 360,
90, 90, 10, 10, 30, 20, 10, 30, 70, 40, 90, 10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
2, 0, 0, 4, 10, 6, 14, 8, 18, 10, 6, 0, 0, 12, 30, 6, 42, 24, 18, 30, 4, 0, 0,
4, 20, 12, 28, 8, 36, 20, 10, 0, 0, 20, 10, 30, 70, 40, 90, 10, 6, 0, 0, 12,
30, 6, 42, 24, 18, 30, 14, 0, 0, 28, 70, 42, 14, 56, 126, 70, 8, 0, 0, 8, 40,
24, 56, 8, 72, 40, 18, 0, 0, 36, 90, 18, 126, 72, 18, 90, 10, 0, 0, 20, 10, 30,
70, 40, 90, 10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 4, 5, 6, 7, 8, 9, 10, 0,
3, 0, 12, 15, 6, 21, 24, 9, 30, 0, 4, 0, 4, 20, 12, 28, 8, 36, 20, 0, 5, 0, 20,
5, 30, 35, 40, 45, 10, 0, 6, 0, 12, 30, 6, 42, 24, 18, 30, 0, 7, 0, 28, 35, 42,
7, 56, 63, 70, 0, 8, 0, 8, 40, 24, 56, 8, 72, 40, 0, 9, 0, 36, 45, 18, 63, 72,
9, 90, 0, 10, 0, 20, 10, 30, 70, 40, 90, 10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6,
3, 0, 12, 30, 6, 42, 24, 18, 30, 6, 3, 0, 12, 30, 6, 42, 24, 18, 30, 12, 12, 0,
12, 60, 12, 84, 24, 36, 60, 30, 15, 0, 60, 30, 30, 210, 120, 90, 30, 6, 6, 0,
12, 30, 6, 42, 24, 18, 30, 42, 21, 0, 84, 210, 42, 42, 168, 126, 210, 24, 24,
0, 24, 120, 24, 168, 24, 72, 120, 18, 9, 0, 36, 90, 18, 126, 72, 18, 90, 30,
30, 0, 60, 30, 30, 210, 120, 90, 30, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 4, 0, 2,
20, 12, 28, 8, 36, 20, 12, 12, 0, 6, 60, 12, 84, 24, 36, 60, 4, 4, 0, 4, 20,
12, 28, 8, 36, 20, 20, 20, 0, 10, 20, 60, 140, 40, 180, 20, 12, 12, 0, 6, 60,
12, 84, 24, 36, 60, 28, 28, 0, 14, 140, 84, 28, 56, 252, 140, 8, 8, 0, 8, 40,
24, 56, 8, 72, 40, 36, 36, 0, 18, 180, 36, 252, 72, 36, 180, 20, 20, 0, 10, 20,
60, 140, 40, 180, 20, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 10, 5, 0, 20, 10, 30, 70,
40, 90, 10, 30, 15, 0, 60, 30, 30, 210, 120, 90, 30, 20, 20, 0, 20, 20, 60,
140, 40, 180, 20, 10, 5, 0, 20, 10, 30, 70, 40, 90, 10, 30, 30, 0, 60, 30, 30,
210, 120, 90, 30, 70, 35, 0, 140, 70, 210, 70, 280, 630, 70, 40, 40, 0, 40, 40,
120, 280, 40, 360, 40, 90, 45, 0, 180, 90, 90, 630, 360, 90, 90, 10, 10, 0, 20,
10, 30, 70, 40, 90, 10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 6, 0, 12, 30, 6, 42,
24, 18, 30, 6, 6, 0, 12, 30, 6, 42, 24, 18, 30, 12, 12, 0, 12, 60, 12, 84, 24,
36, 60, 30, 30, 0, 60, 30, 30, 210, 120, 90, 30, 6, 6, 0, 12, 30, 6, 42, 24,
18, 30, 42, 42, 0, 84, 210, 42, 42, 168, 126, 210, 24, 24, 0, 24, 120, 24, 168,
24, 72, 120, 18, 18, 0, 36, 90, 18, 126, 72, 18, 90, 30, 30, 0, 60, 30, 30,
210, 120, 90, 30, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 14, 7, 0, 28, 70, 42, 14, 56,
126, 70, 42, 21, 0, 84, 210, 42, 42, 168, 126, 210, 28, 28, 0, 28, 140, 84, 28,
56, 252, 140, 70, 35, 0, 140, 70, 210, 70, 280, 630, 70, 42, 42, 0, 84, 210,
42, 42, 168, 126, 210, 14, 7, 0, 28, 70, 42, 14, 56, 126, 70, 56, 56, 0, 56,
280, 168, 56, 56, 504, 280, 126, 63, 0, 252, 630, 126, 126, 504, 126, 630, 70,
70, 0, 140, 70, 210, 70, 280, 630, 70, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8, 8, 0,
8, 40, 24, 56, 8, 72, 40, 24, 24, 0, 24, 120, 24, 168, 24, 72, 120, 8, 8, 0, 8,
40, 24, 56, 8, 72, 40, 40, 40, 0, 40, 40, 120, 280, 40, 360, 40, 24, 24, 0, 24,
120, 24, 168, 24, 72, 120, 56, 56, 0, 56, 280, 168, 56, 56, 504, 280, 8, 8, 0,
8, 40, 24, 56, 8, 72, 40, 72, 72, 0, 72, 360, 72, 504, 72, 72, 360, 40, 40, 0,
40, 40, 120, 280, 40, 360, 40, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 18, 9, 0, 36, 90,
18, 126, 72, 18, 90, 18, 9, 0, 36, 90, 18, 126, 72, 18, 90, 36, 36, 0, 36, 180,
36, 252, 72, 36, 180, 90, 45, 0, 180, 90, 90, 630, 360, 90, 90, 18, 18, 0, 36,
90, 18, 126, 72, 18, 90, 126, 63, 0, 252, 630, 126, 126, 504, 126, 630, 72, 72,
0, 72, 360, 72, 504, 72, 72, 360, 18, 9, 0, 36, 90, 18, 126, 72, 18, 90, 90,
90, 0, 180, 90, 90, 630, 360, 90, 90, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 10, 10, 0,
20, 10, 30, 70, 40, 90, 10, 30, 30, 0, 60, 30, 30, 210, 120, 90, 30, 20, 20, 0,
20, 20, 60, 140, 40, 180, 20, 10, 10, 0, 20, 10, 30, 70, 40, 90, 10, 30, 30, 0,
60, 30, 30, 210, 120, 90, 30, 70, 70, 0, 140, 70, 210, 70, 280, 630, 70, 40,
40, 0, 40, 40, 120, 280, 40, 360, 40, 90, 90, 0, 180, 90, 90, 630, 360, 90, 90,
10, 10, 0, 20, 10, 30, 70, 40, 90, 10, 0, 0, 3, 12, 15, 6, 21, 24, 9, 30, 0, 0,
6, 12, 30, 6, 42, 24, 18, 30, 0, 0, 3, 12, 15, 6, 21, 24, 9, 30, 0, 0, 12, 12,
60, 12, 84, 24, 36, 60, 0, 0, 15, 60, 15, 30, 105, 120, 45, 30, 0, 0, 6, 12, 30,
6, 42, 24, 18, 30, 0, 0, 21, 84, 105, 42, 21, 168, 63, 210, 0, 0, 24, 24, 120,
24, 168, 24, 72, 120, 0, 0, 9, 36, 45, 18, 63, 72, 9, 90, 0, 0, 30, 60, 30, 30,
210, 120, 90, 30, 0, 0, 6, 12, 30, 6, 42, 24, 18, 30, 0, 0, 3, 12, 15, 6, 21,
24, 9, 30, 0, 0, 6, 12, 30, 6, 42, 24, 18, 30, 0, 0, 12, 12, 60, 12, 84, 24,
36, 60, 0, 0, 30, 60, 30, 30, 210, 120, 90, 30, 0, 0, 6, 12, 30, 6, 42, 24, 18,
30, 0, 0, 42, 84, 210, 42, 42, 168, 126, 210, 0, 0, 24, 24, 120, 24, 168, 24,
72, 120, 0, 0, 18, 36, 90, 18, 126, 72, 18, 90, 0, 0, 30, 60, 30, 30, 210, 120,
90, 30, 0, 0, 3, 12, 15, 6, 21, 24, 9, 30, 0, 0, 6, 12, 30, 6, 42, 24, 18, 30,
0, 0, 1, 4, 5, 6, 7, 8, 9, 10, 0, 0, 12, 12, 60, 12, 84, 24, 36, 60, 0, 0, 15,
60, 15, 30, 105, 120, 45, 30, 0, 0, 6, 12, 30, 6, 42, 24, 18, 30, 0, 0, 21, 84,
105, 42, 21, 168, 63, 210, 0, 0, 24, 24, 120, 24, 168, 24, 72, 120, 0, 0, 9,
36, 45, 18, 63, 72, 9, 90, 0, 0, 30, 60, 30, 30, 210, 120, 90, 30, 0, 0, 12, 6,
60, 12, 84, 24, 36, 60, 0, 0, 12, 6, 60, 12, 84, 24, 36, 60, 0, 0, 12, 6, 60,
12, 84, 24, 36, 60, 0, 0, 12, 12, 60, 12, 84, 24, 36, 60, 0, 0, 60, 30, 60, 60,
420, 120, 180, 60, 0, 0, 12, 6, 60, 12, 84, 24, 36, 60, 0, 0, 84, 42, 420, 84,
84, 168, 252, 420, 0, 0, 24, 24, 120, 24, 168, 24, 72, 120, 0, 0, 36, 18, 180,
36, 252, 72, 36, 180, 0, 0, 60, 30, 60, 60, 420, 120, 180, 60, 0, 0, 15, 60,
15, 30, 105, 120, 45, 30, 0, 0, 30, 60, 30, 30, 210, 120, 90, 30, 0, 0, 15, 60,
15, 30, 105, 120, 45, 30, 0, 0, 60, 60, 60, 60, 420, 120, 180, 60, 0, 0, 15,
60, 15, 30, 105, 120, 45, 30, 0, 0, 30, 60, 30, 30, 210, 120, 90, 30, 0, 0,
105, 420, 105, 210, 105, 840, 315, 210, 0, 0, 120, 120, 120, 120, 840, 120,
360, 120, 0, 0, 45, 180, 45, 90, 315, 360, 45, 90, 0, 0, 30, 60, 30, 30, 210,
120, 90, 30, 0, 0, 6, 12, 30, 6, 42, 24, 18, 30, 0, 0, 6, 12, 30, 6, 42, 24,
18, 30, 0, 0, 6, 12, 30, 6, 42, 24, 18, 30, 0, 0, 12, 12, 60, 12, 84, 24, 36,
60, 0, 0, 30, 60, 30, 30, 210, 120, 90, 30, 0, 0, 6, 12, 30, 6, 42, 24, 18, 30,
0, 0, 42, 84, 210, 42, 42, 168, 126, 210, 0, 0, 24, 24, 120, 24, 168, 24, 72,
120, 0, 0, 18, 36, 90, 18, 126, 72, 18, 90, 0, 0, 30, 60, 30, 30, 210, 120, 90,
30, 0, 0, 21, 84, 105, 42, 21, 168, 63, 210, 0, 0, 42, 84, 210, 42, 42, 168,
126, 210, 0, 0, 21, 84, 105, 42, 21, 168, 63, 210, 0, 0, 84, 84, 420, 84, 84,
168, 252, 420, 0, 0, 105, 420, 105, 210, 105, 840, 315, 210, 0, 0, 42, 84, 210,
42, 42, 168, 126, 210, 0, 0, 21, 84, 105, 42, 21, 168, 63, 210, 0, 0, 168, 168,
840, 168, 168, 168, 504, 840, 0, 0, 63, 252, 315, 126, 63, 504, 63, 630, 0, 0,
210, 420, 210, 210, 210, 840, 630, 210, 0, 0, 24, 24, 120, 24, 168, 24, 72,
120, 0, 0, 24, 24, 120, 24, 168, 24, 72, 120, 0, 0, 24, 24, 120, 24, 168, 24,
72, 120, 0, 0, 24, 24, 120, 24, 168, 24, 72, 120, 0, 0, 120, 120, 120, 120,
840, 120, 360, 120, 0, 0, 24, 24, 120, 24, 168, 24, 72, 120, 0, 0, 168, 168,
840, 168, 168, 168, 504, 840, 0, 0, 24, 24, 120, 24, 168, 24, 72, 120, 0, 0,
72, 72, 360, 72, 504, 72, 72, 360, 0, 0, 120, 120, 120, 120, 840, 120, 360,
120, 0, 0, 9, 36, 45, 18, 63, 72, 9, 90, 0, 0, 18, 36, 90, 18, 126, 72, 18, 90,
0, 0, 9, 36, 45, 18, 63, 72, 9, 90, 0, 0, 36, 36, 180, 36, 252, 72, 36, 180, 0,
0, 45, 180, 45, 90, 315, 360, 45, 90, 0, 0, 18, 36, 90, 18, 126, 72, 18, 90, 0,
0, 63, 252, 315, 126, 63, 504, 63, 630, 0, 0, 72, 72, 360, 72, 504, 72, 72,
360, 0, 0, 9, 36, 45, 18, 63, 72, 9, 90, 0, 0, 90, 180, 90, 90, 630, 360, 90,
90, 0, 0, 30, 60, 30, 30, 210, 120, 90, 30, 0, 0, 30, 60, 30, 30, 210, 120, 90,
30, 0, 0, 30, 60, 30, 30, 210, 120, 90, 30, 0, 0, 60, 60, 60, 60, 420, 120,
180, 60, 0, 0, 30, 60, 30, 30, 210, 120, 90, 30, 0, 0, 30, 60, 30, 30, 210,
120, 90, 30, 0, 0, 210, 420, 210, 210, 210, 840, 630, 210, 0, 0, 120, 120, 120,
120, 840, 120, 360, 120, 0, 0, 90, 180, 90, 90, 630, 360, 90, 90, 0, 0, 30, 60,
30, 30, 210, 120, 90, 30, 4, 0, 12, 4, 20, 12, 28, 8, 36, 20, 4, 0, 12, 4, 20,
12, 28, 8, 36, 20, 12, 0, 12, 12, 60, 12, 84, 24, 36, 60, 2, 0, 6, 4, 10, 6,
14, 8, 18, 10, 20, 0, 60, 20, 20, 60, 140, 40, 180, 20, 12, 0, 12, 12, 60, 12,
84, 24, 36, 60, 28, 0, 84, 28, 140, 84, 28, 56, 252, 140, 8, 0, 24, 8, 40, 24,
56, 8, 72, 40, 36, 0, 36, 36, 180, 36, 252, 72, 36, 180, 20, 0, 60, 20, 20, 60,
140, 40, 180, 20, 0, 4, 12, 4, 20, 12, 28, 8, 36, 20, 0, 4, 12, 4, 20, 12, 28,
8, 36, 20, 0, 12, 12, 12, 60, 12, 84, 24, 36, 60, 0, 2, 6, 4, 10, 6, 14, 8, 18,
10, 0, 20, 60, 20, 20, 60, 140, 40, 180, 20, 0, 12, 12, 12, 60, 12, 84, 24, 36,
60, 0, 28, 84, 28, 140, 84, 28, 56, 252, 140, 0, 8, 24, 8, 40, 24, 56, 8, 72,
40, 0, 36, 36, 36, 180, 36, 252, 72, 36, 180, 0, 20, 60, 20, 20, 60, 140, 40,
180, 20, 12, 12, 12, 12, 60, 12, 84, 24, 36, 60, 12, 12, 12, 12, 60, 12, 84,
24, 36, 60, 12, 12, 4, 12, 60, 12, 84, 24, 36, 60, 6, 6, 6, 12, 30, 6, 42, 24,
18, 30, 60, 60, 60, 60, 60, 60, 420, 120, 180, 60, 12, 12, 12, 12, 60, 12, 84,
24, 36, 60, 84, 84, 84, 84, 420, 84, 84, 168, 252, 420, 24, 24, 24, 24, 120,
24, 168, 24, 72, 120, 36, 36, 36, 36, 180, 36, 252, 72, 36, 180, 60, 60, 60,
60, 60, 60, 420, 120, 180, 60, 4, 4, 12, 4, 20, 12, 28, 8, 36, 20, 4, 4, 12, 4,
20, 12, 28, 8, 36, 20, 12, 12, 12, 12, 60, 12, 84, 24, 36, 60, 4, 4, 12, 1, 20,
12, 28, 8, 36, 20, 20, 20, 60, 20, 20, 60, 140, 40, 180, 20, 12, 12, 12, 12,
60, 12, 84, 24, 36, 60, 28, 28, 84, 28, 140, 84, 28, 56, 252, 140, 8, 8, 24, 8,
40, 24, 56, 8, 72, 40, 36, 36, 36, 36, 180, 36, 252, 72, 36, 180, 20, 20, 60,
20, 20, 60, 140, 40, 180, 20, 20, 20, 60, 20, 20, 60, 140, 40, 180, 20, 20, 20,
60, 20, 20, 60, 140, 40, 180, 20, 60, 60, 60, 60, 60, 60, 420, 120, 180, 60,
10, 10, 30, 20, 10, 30, 70, 40, 90, 10, 20, 20, 60, 20, 20, 60, 140, 40, 180,
20, 60, 60, 60, 60, 60, 60, 420, 120, 180, 60, 140, 140, 420, 140, 140, 420,
140, 280, 1260, 140, 40, 40, 120, 40, 40, 120, 280, 40, 360, 40, 180, 180, 180,
180, 180, 180, 1260, 360, 180, 180, 20, 20, 60, 20, 20, 60, 140, 40, 180, 20,
12, 12, 12, 12, 60, 12, 84, 24, 36, 60, 12, 12, 12, 12, 60, 12, 84, 24, 36, 60,
12, 12, 12, 12, 60, 12, 84, 24, 36, 60, 6, 6, 6, 12, 30, 6, 42, 24, 18, 30, 60,
60, 60, 60, 60, 60, 420, 120, 180, 60, 12, 12, 12, 12, 60, 12, 84, 24, 36, 60,
84, 84, 84, 84, 420, 84, 84, 168, 252, 420, 24, 24, 24, 24, 120, 24, 168, 24,
72, 120, 36, 36, 36, 36, 180, 36, 252, 72, 36, 180, 60, 60, 60, 60, 60, 60,
420, 120, 180, 60, 28, 28, 84, 28, 140, 84, 28, 56, 252, 140, 28, 28, 84, 28,
140, 84, 28, 56, 252, 140, 84, 84, 84, 84, 420, 84, 84, 168, 252, 420, 14, 14,
42, 28, 70, 42, 14, 56, 126, 70, 140, 140, 420, 140, 140, 420, 140, 280, 1260,
140, 84, 84, 84, 84, 420, 84, 84, 168, 252, 420, 28, 28, 84, 28, 140, 84, 28,
56, 252, 140, 56, 56, 168, 56, 280, 168, 56, 56, 504, 280, 252, 252, 252, 252,
1260, 252, 252, 504, 252, 1260, 140, 140, 420, 140, 140, 420, 140, 280, 1260,
140, 8, 8, 24, 8, 40, 24, 56, 8, 72, 40, 8, 8, 24, 8, 40, 24, 56, 8, 72, 40,
24, 24, 24, 24, 120, 24, 168, 24, 72, 120, 8, 8, 24, 8, 40, 24, 56, 8, 72, 40,
40, 40, 120, 40, 40, 120, 280, 40, 360, 40, 24, 24, 24, 24, 120, 24, 168, 24,
72, 120, 56, 56, 168, 56, 280, 168, 56, 56, 504, 280, 8, 8, 24, 8, 40, 24, 56,
8, 72, 40, 72, 72, 72, 72, 360, 72, 504, 72, 72, 360, 40, 40, 120, 40, 40, 120,
280, 40, 360, 40, 36, 36, 36, 36, 180, 36, 252, 72, 36, 180, 36, 36, 36, 36,
180, 36, 252, 72, 36, 180, 36, 36, 36, 36, 180, 36, 252, 72, 36, 180, 18, 18,
18, 36, 90, 18, 126, 72, 18, 90, 180, 180, 180, 180, 180, 180, 1260, 360, 180,
180, 36, 36, 36, 36, 180, 36, 252, 72, 36, 180, 252, 252, 252, 252, 1260, 252,
252, 504, 252, 1260, 72, 72, 72, 72, 360, 72, 504, 72, 72, 360, 36, 36, 36, 36,
180, 36, 252, 72, 36, 180, 180, 180, 180, 180, 180, 180, 1260, 360, 180, 180,
20, 20, 60, 20, 20, 60, 140, 40, 180, 20, 20, 20, 60, 20, 20, 60, 140, 40, 180,
20, 60, 60, 60, 60, 60, 60, 420, 120, 180, 60, 10, 10, 30, 20, 10, 30, 70, 40,
90, 10, 20, 20, 60, 20, 20, 60, 140, 40, 180, 20, 60, 60, 60, 60, 60, 60, 420,
120, 180, 60, 140, 140, 420, 140, 140, 420, 140, 280, 1260, 140, 40, 40, 120,
40, 40, 120, 280, 40, 360, 40, 180, 180, 180, 180, 180, 180, 1260, 360, 180,
180, 20, 20, 60, 20, 20, 60, 140, 40, 180, 20, 5, 0, 15, 20, 5, 30, 35, 40, 45,
10, 10, 0, 30, 20, 10, 30, 70, 40, 90, 10, 15, 0, 15, 60, 15, 30, 105, 120, 45,
30, 20, 0, 60, 20, 20, 60, 140, 40, 180, 20, 5, 0, 15, 20, 5, 30, 35, 40, 45,
10, 30, 0, 30, 60, 30, 30, 210, 120, 90, 30, 35, 0, 105, 140, 35, 210, 35, 280,
315, 70, 40, 0, 120, 40, 40, 120, 280, 40, 360, 40, 45, 0, 45, 180, 45, 90,
315, 360, 45, 90, 10, 0, 30, 20, 10, 30, 70, 40, 90, 10, 0, 10, 30, 20, 10, 30,
70, 40, 90, 10, 0, 5, 15, 20, 5, 30, 35, 40, 45, 10, 0, 30, 30, 60, 30, 30,
210, 120, 90, 30, 0, 20, 60, 20, 20, 60, 140, 40, 180, 20, 0, 10, 30, 20, 10,
30, 70, 40, 90, 10, 0, 30, 30, 60, 30, 30, 210, 120, 90, 30, 0, 70, 210, 140,
70, 210, 70, 280, 630, 70, 0, 40, 120, 40, 40, 120, 280, 40, 360, 40, 0, 90,
90, 180, 90, 90, 630, 360, 90, 90, 0, 10, 30, 20, 10, 30, 70, 40, 90, 10, 15,
30, 15, 60, 15, 30, 105, 120, 45, 30, 30, 15, 30, 60, 30, 30, 210, 120, 90, 30,
15, 30, 5, 60, 15, 30, 105, 120, 45, 30, 60, 60, 60, 60, 60, 60, 420, 120, 180,
60, 15, 30, 15, 60, 15, 30, 105, 120, 45, 30, 30, 30, 30, 60, 30, 30, 210, 120,
90, 30, 105, 210, 105, 420, 105, 210, 105, 840, 315, 210, 120, 120, 120, 120,
120, 120, 840, 120, 360, 120, 45, 90, 45, 180, 45, 90, 315, 360, 45, 90, 30,
30, 30, 60, 30, 30, 210, 120, 90, 30, 20, 20, 60, 10, 20, 60, 140, 40, 180, 20,
20, 20, 60, 10, 20, 60, 140, 40, 180, 20, 60, 60, 60, 30, 60, 60, 420, 120,
180, 60, 20, 20, 60, 20, 20, 60, 140, 40, 180, 20, 20, 20, 60, 10, 20, 60, 140,
40, 180, 20, 60, 60, 60, 30, 60, 60, 420, 120, 180, 60, 140, 140, 420, 70, 140,
420, 140, 280, 1260, 140, 40, 40, 120, 40, 40, 120, 280, 40, 360, 40, 180, 180,
180, 90, 180, 180, 1260, 360, 180, 180, 20, 20, 60, 10, 20, 60, 140, 40, 180,
20, 5, 10, 15, 20, 5, 30, 35, 40, 45, 10, 10, 5, 30, 20, 10, 30, 70, 40, 90,
10, 15, 30, 15, 60, 15, 30, 105, 120, 45, 30, 20, 20, 60, 20, 20, 60, 140, 40,
180, 20, 5, 10, 15, 20, 5, 30, 35, 40, 45, 10, 30, 30, 30, 60, 30, 30, 210,
120, 90, 30, 35, 70, 105, 140, 35, 210, 35, 280, 315, 70, 40, 40, 120, 40, 40,
120, 280, 40, 360, 40, 45, 90, 45, 180, 45, 90, 315, 360, 45, 90, 10, 10, 30,
20, 10, 30, 70, 40, 90, 10, 30, 30, 30, 60, 30, 30, 210, 120, 90, 30, 30, 30,
30, 60, 30, 30, 210, 120, 90, 30, 30, 30, 30, 60, 30, 30, 210, 120, 90, 30, 60,
60, 60, 60, 60, 60, 420, 120, 180, 60, 30, 30, 30, 60, 30, 30, 210, 120, 90,
30, 30, 30, 30, 60, 30, 30, 210, 120, 90, 30, 210, 210, 210, 420, 210, 210,
210, 840, 630, 210, 120, 120, 120, 120, 120, 120, 840, 120, 360, 120, 90, 90,
90, 180, 90, 90, 630, 360, 90, 90, 30, 30, 30, 60, 30, 30, 210, 120, 90, 30,
35, 70, 105, 140, 35, 210, 35, 280, 315, 70, 70, 35, 210, 140, 70, 210, 70,
280, 630, 70, 105, 210, 105, 420, 105, 210, 105, 840, 315, 210, 140, 140, 420,
140, 140, 420, 140, 280, 1260, 140, 35, 70, 105, 140, 35, 210, 35, 280, 315,
70, 210, 210, 210, 420, 210, 210, 210, 840, 630, 210, 35, 70, 105, 140, 35,
210, 35, 280, 315, 70, 280, 280, 840, 280, 280, 840, 280, 280, 2520, 280, 315,
630, 315, 1260, 315, 630, 315, 2520, 315, 630, 70, 70, 210, 140, 70, 210, 70,
280, 630, 70, 40, 40, 120, 40, 40, 120, 280, 40, 360, 40, 40, 40, 120, 40, 40,
120, 280, 40, 360, 40, 120, 120, 120, 120, 120, 120, 840, 120, 360, 120, 40,
40, 120, 40, 40, 120, 280, 40, 360, 40, 40, 40, 120, 40, 40, 120, 280, 40, 360,
40, 120, 120, 120, 120, 120, 120, 840, 120, 360, 120, 280, 280, 840, 280, 280,
840, 280, 280, 2520, 280, 40, 40, 120, 40, 40, 120, 280, 40, 360, 40, 360, 360,
360, 360, 360, 360, 2520, 360, 360, 360, 40, 40, 120, 40, 40, 120, 280, 40,
360, 40, 45, 90, 45, 180, 45, 90, 315, 360, 45, 90, 90, 45, 90, 180, 90, 90,
630, 360, 90, 90, 45, 90, 45, 180, 45, 90, 315, 360, 45, 90, 180, 180, 180,
180, 180, 180, 1260, 360, 180, 180, 45, 90, 45, 180, 45, 90, 315, 360, 45, 90,
90, 90, 90, 180, 90, 90, 630, 360, 90, 90, 315, 630, 315, 1260, 315, 630, 315,
2520, 315, 630, 360, 360, 360, 360, 360, 360, 2520, 360, 360, 360, 45, 90, 45,
180, 45, 90, 315, 360, 45, 90, 90, 90, 90, 180, 90, 90, 630, 360, 90, 90, 10,
10, 30, 20, 10, 30, 70, 40, 90, 10, 10, 10, 30, 20, 10, 30, 70, 40, 90, 10, 30,
30, 30, 60, 30, 30, 210, 120, 90, 30, 20, 20, 60, 20, 20, 60, 140, 40, 180, 20,
10, 10, 30, 20, 10, 30, 70, 40, 90, 10, 30, 30, 30, 60, 30, 30, 210, 120, 90,
30, 70, 70, 210, 140, 70, 210, 70, 280, 630, 70, 40, 40, 120, 40, 40, 120, 280,
40, 360, 40, 90, 90, 90, 180, 90, 90, 630, 360, 90, 90, 10, 10, 30, 20, 10, 30,
70, 40, 90, 10, 6, 0, 6, 12, 30, 3, 42, 24, 18, 30, 6, 0, 6, 12, 30, 6, 42, 24,
18, 30, 6, 0, 6, 12, 30, 3, 42, 24, 18, 30, 12, 0, 12, 12, 60, 12, 84, 24, 36,
60, 30, 0, 30, 60, 30, 15, 210, 120, 90, 30, 6, 0, 6, 12, 30, 6, 42, 24, 18,
30, 42, 0, 42, 84, 210, 21, 42, 168, 126, 210, 24, 0, 24, 24, 120, 24, 168, 24,
72, 120, 18, 0, 18, 36, 90, 9, 126, 72, 18, 90, 30, 0, 30, 60, 30, 30, 210,
120, 90, 30, 0, 6, 6, 12, 30, 6, 42, 24, 18, 30, 0, 6, 6, 12, 30, 3, 42, 24,
18, 30, 0, 6, 6, 12, 30, 6, 42, 24, 18, 30, 0, 12, 12, 12, 60, 12, 84, 24, 36,
60, 0, 30, 30, 60, 30, 30, 210, 120, 90, 30, 0, 6, 6, 12, 30, 6, 42, 24, 18,
30, 0, 42, 42, 84, 210, 42, 42, 168, 126, 210, 0, 24, 24, 24, 120, 24, 168, 24,
72, 120, 0, 18, 18, 36, 90, 18, 126, 72, 18, 90, 0, 30, 30, 60, 30, 30, 210,
120, 90, 30, 6, 6, 6, 12, 30, 3, 42, 24, 18, 30, 6, 6, 6, 12, 30, 6, 42, 24,
18, 30, 6, 6, 6, 12, 30, 3, 42, 24, 18, 30, 12, 12, 12, 12, 60, 12, 84, 24, 36,
60, 30, 30, 30, 60, 30, 15, 210, 120, 90, 30, 6, 6, 6, 12, 30, 6, 42, 24, 18,
30, 42, 42, 42, 84, 210, 21, 42, 168, 126, 210, 24, 24, 24, 24, 120, 24, 168,
24, 72, 120, 18, 18, 18, 36, 90, 9, 126, 72, 18, 90, 30, 30, 30, 60, 30, 30,
210, 120, 90, 30, 12, 12, 12, 6, 60, 12, 84, 24, 36, 60, 12, 12, 12, 6, 60, 12,
84, 24, 36, 60, 12, 12, 12, 6, 60, 12, 84, 24, 36, 60, 12, 12, 12, 12, 60, 12,
84, 24, 36, 60, 60, 60, 60, 30, 60, 60, 420, 120, 180, 60, 12, 12, 12, 6, 60,
12, 84, 24, 36, 60, 84, 84, 84, 42, 420, 84, 84, 168, 252, 420, 24, 24, 24, 24,
120, 24, 168, 24, 72, 120, 36, 36, 36, 18, 180, 36, 252, 72, 36, 180, 60, 60,
60, 30, 60, 60, 420, 120, 180, 60, 30, 30, 30, 60, 30, 15, 210, 120, 90, 30,
30, 30, 30, 60, 30, 30, 210, 120, 90, 30, 30, 30, 30, 60, 30, 15, 210, 120, 90,
30, 60, 60, 60, 60, 60, 60, 420, 120, 180, 60, 30, 30, 30, 60, 30, 15, 210,
120, 90, 30, 30, 30, 30, 60, 30, 30, 210, 120, 90, 30, 210, 210, 210, 420, 210,
105, 210, 840, 630, 210, 120, 120, 120, 120, 120, 120, 840, 120, 360, 120, 90,
90, 90, 180, 90, 45, 630, 360, 90, 90, 30, 30, 30, 60, 30, 30, 210, 120, 90,
30, 6, 6, 6, 12, 30, 6, 42, 24, 18, 30, 6, 6, 6, 12, 30, 6, 42, 24, 18, 30, 6,
6, 6, 12, 30, 6, 42, 24, 18, 30, 12, 12, 12, 12, 60, 12, 84, 24, 36, 60, 30,
30, 30, 60, 30, 30, 210, 120, 90, 30, 6, 6, 6, 12, 30, 6, 42, 24, 18, 30, 42,
42, 42, 84, 210, 42, 42, 168, 126, 210, 24, 24, 24, 24, 120, 24, 168, 24, 72,
120, 18, 18, 18, 36, 90, 18, 126, 72, 18, 90, 30, 30, 30, 60, 30, 30, 210, 120,
90, 30, 42, 42, 42, 84, 210, 21, 42, 168, 126, 210, 42, 42, 42, 84, 210, 42,
42, 168, 126, 210, 42, 42, 42, 84, 210, 21, 42, 168, 126, 210, 84, 84, 84, 84,
420, 84, 84, 168, 252, 420, 210, 210, 210, 420, 210, 105, 210, 840, 630, 210,
42, 42, 42, 84, 210, 42, 42, 168, 126, 210, 42, 42, 42, 84, 210, 21, 42, 168,
126, 210, 168, 168, 168, 168, 840, 168, 168, 168, 504, 840, 126, 126, 126, 252,
630, 63, 126, 504, 126, 630, 210, 210, 210, 420, 210, 210, 210, 840, 630, 210,
24, 24, 24, 24, 120, 24, 168, 24, 72, 120, 24, 24, 24, 24, 120, 24, 168, 24,
72, 120, 24, 24, 24, 24, 120, 24, 168, 24, 72, 120, 24, 24, 24, 24, 120, 24,
168, 24, 72, 120, 120, 120, 120, 120, 120, 120, 840, 120, 360, 120, 24, 24, 24,
24, 120, 24, 168, 24, 72, 120, 168, 168, 168, 168, 840, 168, 168, 168, 504,
840, 24, 24, 24, 24, 120, 24, 168, 24, 72, 120, 72, 72, 72, 72, 360, 72, 504,
72, 72, 360, 120, 120, 120, 120, 120, 120, 840, 120, 360, 120, 18, 18, 18, 36,
90, 9, 126, 72, 18, 90, 18, 18, 18, 36, 90, 18, 126, 72, 18, 90, 18, 18, 18,
36, 90, 9, 126, 72, 18, 90, 36, 36, 36, 36, 180, 36, 252, 72, 36, 180, 90, 90,
90, 180, 90, 45, 630, 360, 90, 90, 18, 18, 18, 36, 90, 18, 126, 72, 18, 90,
126, 126, 126, 252, 630, 63, 126, 504, 126, 630, 72, 72, 72, 72, 360, 72, 504,
72, 72, 360, 18, 18, 18, 36, 90, 9, 126, 72, 18, 90, 90, 90, 90, 180, 90, 90,
630, 360, 90, 90, 30, 30, 30, 60, 30, 30, 210, 120, 90, 30, 30, 30, 30, 60, 30,
30, 210, 120, 90, 30, 30, 30, 30, 60, 30, 30, 210, 120, 90, 30, 60, 60, 60, 60,
60, 60, 420, 120, 180, 60, 30, 30, 30, 60, 30, 30, 210, 120, 90, 30, 30, 30,
30, 60, 30, 30, 210, 120, 90, 30, 210, 210, 210, 420, 210, 210, 210, 840, 630,
210, 120, 120, 120, 120, 120, 120, 840, 120, 360, 120, 90, 90, 90, 180, 90, 90,
630, 360, 90, 90, 30, 30, 30, 60, 30, 30, 210, 120, 90, 30, 7, 0, 21, 28, 35,
42, 7, 56, 63, 70, 14, 0, 42, 28, 70, 42, 14, 56, 126, 70, 21, 0, 21, 84, 105,
42, 21, 168, 63, 210, 28, 0, 84, 28, 140, 84, 28, 56, 252, 140, 35, 0, 105,
140, 35, 210, 35, 280, 315, 70, 42, 0, 42, 84, 210, 42, 42, 168, 126, 210, 7,
0, 21, 28, 35, 42, 7, 56, 63, 70, 56, 0, 168, 56, 280, 168, 56, 56, 504, 280,
63, 0, 63, 252, 315, 126, 63, 504, 63, 630, 70, 0, 210, 140, 70, 210, 70, 280,
630, 70, 0, 14, 42, 28, 70, 42, 14, 56, 126, 70, 0, 7, 21, 28, 35, 42, 7, 56,
63, 70, 0, 42, 42, 84, 210, 42, 42, 168, 126, 210, 0, 28, 84, 28, 140, 84, 28,
56, 252, 140, 0, 70, 210, 140, 70, 210, 70, 280, 630, 70, 0, 42, 42, 84, 210,
42, 42, 168, 126, 210, 0, 14, 42, 28, 70, 42, 14, 56, 126, 70, 0, 56, 168, 56,
280, 168, 56, 56, 504, 280, 0, 126, 126, 252, 630, 126, 126, 504, 126, 630, 0,
70, 210, 140, 70, 210, 70, 280, 630, 70, 21, 42, 21, 84, 105, 42, 21, 168, 63, 210,
42, 21, 42, 84, 210, 42, 42, 168, 126, 210, 21, 42, 7, 84, 105, 42, 21, 168,
63, 210, 84, 84, 84, 84, 420, 84, 84, 168, 252, 420, 105, 210, 105, 420, 105,
210, 105, 840, 315, 210, 42, 42, 42, 84, 210, 42, 42, 168, 126, 210, 21, 42,
21, 84, 105, 42, 21, 168, 63, 210, 168, 168, 168, 168, 840, 168, 168, 168, 504,
840, 63, 126, 63, 252, 315, 126, 63, 504, 63, 630, 210, 210, 210, 420, 210,
210, 210, 840, 630, 210, 28, 28, 84, 14, 140, 84, 28, 56, 252, 140, 28, 28, 84,
14, 140, 84, 28, 56, 252, 140, 84, 84, 84, 42, 420, 84, 84, 168, 252, 420, 28,
28, 84, 28, 140, 84, 28, 56, 252, 140, 140, 140, 420, 70, 140, 420, 140, 280,
1260, 140, 84, 84, 84, 42, 420, 84, 84, 168, 252, 420, 28, 28, 84, 14, 140, 84,
28, 56, 252, 140, 56, 56, 168, 56, 280, 168, 56, 56, 504, 280, 252, 252, 252,
126, 1260, 252, 252, 504, 252, 1260, 140, 140, 420, 70, 140, 420, 140, 280,
1260, 140, 35, 70, 105, 140, 35, 210, 35, 280, 315, 70, 70, 35, 210, 140, 70,
210, 70, 280, 630, 70, 105, 210, 105, 420, 105, 210, 105, 840, 315, 210, 140,
140, 420, 140, 140, 420, 140, 280, 1260, 140, 35, 70, 105, 140, 35, 210, 35,
280, 315, 70, 210, 210, 210, 420, 210, 210, 210, 840, 630, 210, 35, 70, 105,
140, 35, 210, 35, 280, 315, 70, 280, 280, 840, 280, 280, 840, 280, 280, 2520,
280, 315, 630, 315, 1260, 315, 630, 315, 2520, 315, 630, 70, 70, 210, 140, 70,
210, 70, 280, 630, 70, 42, 42, 42, 84, 210, 42, 42, 168, 126, 210, 42, 42, 42,
84, 210, 42, 42, 168, 126, 210, 42, 42, 42, 84, 210, 42, 42, 168, 126, 210, 84,
84, 84, 84, 420, 84, 84, 168, 252, 420, 210, 210, 210, 420, 210, 210, 210, 840,
630, 210, 42, 42, 42, 84, 210, 42, 42, 168, 126, 210, 42, 42, 42, 84, 210, 42,
42, 168, 126, 210, 168, 168, 168, 168, 840, 168, 168, 168, 504, 840, 126, 126,
126, 252, 630, 126, 126, 504, 126, 630, 210, 210, 210, 420, 210, 210, 210, 840,
630, 210, 7, 14, 21, 28, 35, 42, 7, 56, 63, 70, 14, 7, 42, 28, 70, 42, 14, 56,
126, 70, 21, 42, 21, 84, 105, 42, 21, 168, 63, 210, 28, 28, 84, 28, 140, 84,
28, 56, 252, 140, 35, 70, 105, 140, 35, 210, 35, 280, 315, 70, 42, 42, 42, 84,
210, 42, 42, 168, 126, 210, 7, 14, 21, 28, 35, 42, 7, 56, 63, 70, 56, 56, 168,
56, 280, 168, 56, 56, 504, 280, 63, 126, 63, 252, 315, 126, 63, 504, 63, 630,
70, 70, 210, 140, 70, 210, 70, 280, 630, 70, 56, 56, 168, 56, 280, 168, 56, 56,
504, 280, 56, 56, 168, 56, 280, 168, 56, 56, 504, 280, 168, 168, 168, 168, 840,
168, 168, 168, 504, 840, 56, 56, 168, 56, 280, 168, 56, 56, 504, 280, 280, 280,
840, 280, 280, 840, 280, 280, 2520, 280, 168, 168, 168, 168, 840, 168, 168,
168, 504, 840, 56, 56, 168, 56, 280, 168, 56, 56, 504, 280, 56, 56, 168, 56,
280, 168, 56, 56, 504, 280, 504, 504, 504, 504, 2520, 504, 504, 504, 504, 2520,
280, 280, 840, 280, 280, 840, 280, 280, 2520, 280, 63, 126, 63, 252, 315, 126,
63, 504, 63, 630, 126, 63, 126, 252, 630, 126, 126, 504, 126, 630, 63, 126, 63,
252, 315, 126, 63, 504, 63, 630, 252, 252, 252, 252, 1260, 252, 252, 504, 252,
1260, 315, 630, 315, 1260, 315, 630, 315, 2520, 315, 630, 126, 126, 126, 252,
630, 126, 126, 504, 126, 630, 63, 126, 63, 252, 315, 126, 63, 504, 63, 630,
504, 504, 504, 504, 2520, 504, 504, 504, 504, 2520, 63, 126, 63, 252, 315, 126,
63, 504, 63, 630, 630, 630, 630, 1260, 630, 630, 630, 2520, 630, 630, 70, 70,
210, 140, 70, 210, 70, 280, 630, 70, 70, 70, 210, 140, 70, 210, 70, 280, 630,
70, 210, 210, 210, 420, 210, 210, 210, 840, 630, 210, 140, 140, 420, 140, 140,
420, 140, 280, 1260, 140, 70, 70, 210, 140, 70, 210, 70, 280, 630, 70, 210,
210, 210, 420, 210, 210, 210, 840, 630, 210, 70, 70, 210, 140, 70, 210, 70,
280, 630, 70, 280, 280, 840, 280, 280, 840, 280, 280, 2520, 280, 630, 630, 630,
1260, 630, 630, 630, 2520, 630, 630, 70, 70, 210, 140, 70, 210, 70, 280, 630,
70, 8, 0, 24, 8, 40, 24, 56, 8, 72, 40, 8, 0, 24, 8, 40, 24, 56, 8, 72, 40, 24,
0, 24, 24, 120, 24, 168, 24, 72, 120, 8, 0, 24, 8, 40, 24, 56, 8, 72, 40, 40,
0, 120, 40, 40, 120, 280, 40, 360, 40, 24, 0, 24, 24, 120, 24, 168, 24, 72,
120, 56, 0, 168, 56, 280, 168, 56, 56, 504, 280, 8, 0, 24, 8, 40, 24, 56, 8,
72, 40, 72, 0, 72, 72, 360, 72, 504, 72, 72, 360, 40, 0, 120, 40, 40, 120, 280,
40, 360, 40, 0, 8, 24, 8, 40, 24, 56, 8, 72, 40, 0, 8, 24, 8, 40, 24, 56, 8,
72, 40, 0, 24, 24, 24, 120, 24, 168, 24, 72, 120, 0, 8, 24, 8, 40, 24, 56, 8,
72, 40, 0, 40, 120, 40, 40, 120, 280, 40, 360, 40, 0, 24, 24, 24, 120, 24, 168,
24, 72, 120, 0, 56, 168, 56, 280, 168, 56, 56, 504, 280, 0, 8, 24, 8, 40, 24,
56, 8, 72, 40, 0, 72, 72, 72, 360, 72, 504, 72, 72, 360, 0, 40, 120, 40, 40,
120, 280, 40, 360, 40, 24, 24, 24, 24, 120, 24, 168, 24, 72, 120, 24, 24, 24,
24, 120, 24, 168, 24, 72, 120, 24, 24, 8, 24, 120, 24, 168, 24, 72, 120, 24,
24, 24, 24, 120, 24, 168, 24, 72, 120, 120, 120, 120, 120, 120, 120, 840, 120,
360, 120, 24, 24, 24, 24, 120, 24, 168, 24, 72, 120, 168, 168, 168, 168, 840,
168, 168, 168, 504, 840, 24, 24, 24, 24, 120, 24, 168, 24, 72, 120, 72, 72, 72,
72, 360, 72, 504, 72, 72, 360, 120, 120, 120, 120, 120, 120, 840, 120, 360,
120, 8, 8, 24, 8, 40, 24, 56, 8, 72, 40, 8, 8, 24, 8, 40, 24, 56, 8, 72, 40,
24, 24, 24, 24, 120, 24, 168, 24, 72, 120, 8, 8, 24, 8, 40, 24, 56, 8, 72, 40,
40, 40, 120, 40, 40, 120, 280, 40, 360, 40, 24, 24, 24, 24, 120, 24, 168, 24,
72, 120, 56, 56, 168, 56, 280, 168, 56, 56, 504, 280, 8, 8, 24, 8, 40, 24, 56,
8, 72, 40, 72, 72, 72, 72, 360, 72, 504, 72, 72, 360, 40, 40, 120, 40, 40, 120,
280, 40, 360, 40, 40, 40, 120, 40, 40, 120, 280, 40, 360, 40, 40, 40, 120, 40,
40, 120, 280, 40, 360, 40, 120, 120, 120, 120, 120, 120, 840, 120, 360, 120,
40, 40, 120, 40, 40, 120, 280, 40, 360, 40, 40, 40, 120, 40, 40, 120, 280, 40,
360, 40, 120, 120, 120, 120, 120, 120, 840, 120, 360, 120, 280, 280, 840, 280,
280, 840, 280, 280, 2520, 280, 40, 40, 120, 40, 40, 120, 280, 40, 360, 40, 360,
360, 360, 360, 360, 360, 2520, 360, 360, 360, 40, 40, 120, 40, 40, 120, 280,
40, 360, 40, 24, 24, 24, 24, 120, 24, 168, 24, 72, 120, 24, 24, 24, 24, 120,
24, 168, 24, 72, 120, 24, 24, 24, 24, 120, 24, 168, 24, 72, 120, 24, 24, 24,
24, 120, 24, 168, 24, 72, 120, 120, 120, 120, 120, 120, 120, 840, 120, 360,
120, 24, 24, 24, 24, 120, 24, 168, 24, 72, 120, 168, 168, 168, 168, 840, 168,
168, 168, 504, 840, 24, 24, 24, 24, 120, 24, 168, 24, 72, 120, 72, 72, 72, 72,
360, 72, 504, 72, 72, 360, 120, 120, 120, 120, 120, 120, 840, 120, 360, 120,
56, 56, 168, 56, 280, 168, 56, 56, 504, 280, 56, 56, 168, 56, 280, 168, 56, 56,
504, 280, 168, 168, 168, 168, 840, 168, 168, 168, 504, 840, 56, 56, 168, 56,
280, 168, 56, 56, 504, 280, 280, 280, 840, 280, 280, 840, 280, 280, 2520, 280,
168, 168, 168, 168, 840, 168, 168, 168, 504, 840, 56, 56, 168, 56, 280, 168,
56, 56, 504, 280, 56, 56, 168, 56, 280, 168, 56, 56, 504, 280, 504, 504, 504, 504,
2520, 504, 504, 504, 504, 2520, 280, 280, 840, 280, 280, 840, 280, 280, 2520,
280, 8, 8, 24, 8, 40, 24, 56, 8, 72, 40, 8, 8, 24, 8, 40, 24, 56, 8, 72, 40,
24, 24, 24, 24, 120, 24, 168, 24, 72, 120, 8, 8, 24, 8, 40, 24, 56, 8, 72, 40,
40, 40, 120, 40, 40, 120, 280, 40, 360, 40, 24, 24, 24, 24, 120, 24, 168, 24,
72, 120, 56, 56, 168, 56, 280, 168, 56, 56, 504, 280, 8, 8, 24, 8, 40, 24, 56,
8, 72, 40, 72, 72, 72, 72, 360, 72, 504, 72, 72, 360, 40, 40, 120, 40, 40, 120,
280, 40, 360, 40, 72, 72, 72, 72, 360, 72, 504, 72, 72, 360, 72, 72, 72, 72,
360, 72, 504, 72, 72, 360, 72, 72, 72, 72, 360, 72, 504, 72, 72, 360, 72, 72,
72, 72, 360, 72, 504, 72, 72, 360, 360, 360, 360, 360, 360, 360, 2520, 360,
360, 360, 72, 72, 72, 72, 360, 72, 504, 72, 72, 360, 504, 504, 504, 504, 2520,
504, 504, 504, 504, 2520, 72, 72, 72, 72, 360, 72, 504, 72, 72, 360, 72, 72,
72, 72, 360, 72, 504, 72, 72, 360, 360, 360, 360, 360, 360, 360, 2520, 360,
360, 360, 40, 40, 120, 40, 40, 120, 280, 40, 360, 40, 40, 40, 120, 40, 40, 120,
280, 40, 360, 40, 120, 120, 120, 120, 120, 120, 840, 120, 360, 120, 40, 40,
120, 40, 40, 120, 280, 40, 360, 40, 40, 40, 120, 40, 40, 120, 280, 40, 360, 40,
120, 120, 120, 120, 120, 120, 840, 120, 360, 120, 280, 280, 840, 280, 280, 840,
280, 280, 2520, 280, 40, 40, 120, 40, 40, 120, 280, 40, 360, 40, 360, 360, 360,
360, 360, 360, 2520, 360, 360, 360, 40, 40, 120, 40, 40, 120, 280, 40, 360, 40,
9, 0, 9, 36, 45, 18, 63, 72, 9, 90, 18, 0, 18, 36, 90, 18, 126, 72, 18, 90, 9,
0, 9, 36, 45, 18, 63, 72, 9, 90, 36, 0, 36, 36, 180, 36, 252, 72, 36, 180, 45,
0, 45, 180, 45, 90, 315, 360, 45, 90, 18, 0, 18, 36, 90, 18, 126, 72, 18, 90,
63, 0, 63, 252, 315, 126, 63, 504, 63, 630, 72, 0, 72, 72, 360, 72, 504, 72,
72, 360, 9, 0, 9, 36, 45, 18, 63, 72, 9, 90, 90, 0, 90, 180, 90, 90, 630, 360,
90, 90, 0, 18, 18, 36, 90, 18, 126, 72, 18, 90, 0, 9, 9, 36, 45, 18, 63, 72, 9,
90, 0, 18, 18, 36, 90, 18, 126, 72, 18, 90, 0, 36, 36, 36, 180, 36, 252, 72,
36, 180, 0, 90, 90, 180, 90, 90, 630, 360, 90, 90, 0, 18, 18, 36, 90, 18, 126,
72, 18, 90, 0, 126, 126, 252, 630, 126, 126, 504, 126, 630, 0, 72, 72, 72, 360,
72, 504, 72, 72, 360, 0, 18, 18, 36, 90, 18, 126, 72, 18, 90, 0, 90, 90, 180,
90, 90, 630, 360, 90, 90, 9, 18, 9, 36, 45, 18, 63, 72, 9, 90, 18, 9, 18, 36,
90, 18, 126, 72, 18, 90, 9, 18, 9, 36, 45, 18, 63, 72, 9, 90, 36, 36, 36, 36,
180, 36, 252, 72, 36, 180, 45, 90, 45, 180, 45, 90, 315, 360, 45, 90, 18, 18,
18, 36, 90, 18, 126, 72, 18, 90, 63, 126, 63, 252, 315, 126, 63, 504, 63, 630,
72, 72, 72, 72, 360, 72, 504, 72, 72, 360, 9, 18, 9, 36, 45, 18, 63, 72, 9, 90,
90, 90, 90, 180, 90, 90, 630, 360, 90, 90, 36, 36, 36, 18, 180, 36, 252, 72,
36, 180, 36, 36, 36, 18, 180, 36, 252, 72, 36, 180, 36, 36, 36, 18, 180, 36,
252, 72, 36, 180, 36, 36, 36, 36, 180, 36, 252, 72, 36, 180, 180, 180, 180, 90,
180, 180, 1260, 360, 180, 180, 36, 36, 36, 18, 180, 36, 252, 72, 36, 180, 252,
252, 252, 126, 1260, 252, 252, 504, 252, 1260, 72, 72, 72, 72, 360, 72, 504,
72, 72, 360, 36, 36, 36, 18, 180, 36, 252, 72, 36, 180, 180, 180, 180, 90, 180,
180, 1260, 360, 180, 180, 45, 90, 45, 180, 45, 90, 315, 360, 45, 90, 90, 45,
90, 180, 90, 90, 630, 360, 90, 90, 45, 90, 45, 180, 45, 90, 315, 360, 45, 90,
180, 180, 180, 180, 180, 180, 1260, 360, 180, 180, 45, 90, 45, 180, 45, 90,
315, 360, 45, 90, 90, 90, 90, 180, 90, 90, 630, 360, 90, 90, 315, 630, 315,
1260, 315, 630, 315, 2520, 315, 630, 360, 360, 360, 360, 360, 360, 2520, 360,
360, 360, 45, 90, 45, 180, 45, 90, 315, 360, 45, 90, 90, 90, 90, 180, 90, 90,
630, 360, 90, 90, 18, 18, 18, 36, 90, 18, 126, 72, 18, 90, 18, 18, 18, 36, 90,
18, 126, 72, 18, 90, 18, 18, 18, 36, 90, 18, 126, 72, 18, 90, 36, 36, 36, 36,
180, 36, 252, 72, 36, 180, 90, 90, 90, 180, 90, 90, 630, 360, 90, 90, 18, 18,
18, 36, 90, 18, 126, 72, 18, 90, 126, 126, 126, 252, 630, 126, 126, 504, 126, 630,
72, 72, 72, 72, 360, 72, 504, 72, 72, 360, 18, 18, 18, 36, 90, 18, 126, 72, 18,
90, 90, 90, 90, 180, 90, 90, 630, 360, 90, 90, 63, 126, 63, 252, 315, 126, 63,
504, 63, 630, 126, 63, 126, 252, 630, 126, 126, 504, 126, 630, 63, 126, 63,
252, 315, 126, 63, 504, 63, 630, 252, 252, 252, 252, 1260, 252, 252, 504, 252,
1260, 315, 630, 315, 1260, 315, 630, 315, 2520, 315, 630, 126, 126, 126, 252,
630, 126, 126, 504, 126, 630, 63, 126, 63, 252, 315, 126, 63, 504, 63, 630,
504, 504, 504, 504, 2520, 504, 504, 504, 504, 2520, 63, 126, 63, 252, 315, 126,
63, 504, 63, 630, 630, 630, 630, 1260, 630, 630, 630, 2520, 630, 630, 72, 72,
72, 72, 360, 72, 504, 72, 72, 360, 72, 72, 72, 72, 360, 72, 504, 72, 72, 360,
72, 72, 72, 72, 360, 72, 504, 72, 72, 360, 72, 72, 72, 72, 360, 72, 504, 72,
72, 360, 360, 360, 360, 360, 360, 360, 2520, 360, 360, 360, 72, 72, 72, 72,
360, 72, 504, 72, 72, 360, 504, 504, 504, 504, 2520, 504, 504, 504, 504, 2520,
72, 72, 72, 72, 360, 72, 504, 72, 72, 360, 72, 72, 72, 72, 360, 72, 504, 72,
72, 360, 360, 360, 360, 360, 360, 360, 2520, 360, 360, 360, 9, 18, 9, 36, 45,
18, 63, 72, 9, 90, 18, 9, 18, 36, 90, 18, 126, 72, 18, 90, 9, 18, 9, 36, 45,
18, 63, 72, 9, 90, 36, 36, 36, 36, 180, 36, 252, 72, 36, 180, 45, 90, 45, 180,
45, 90, 315, 360, 45, 90, 18, 18, 18, 36, 90, 18, 126, 72, 18, 90, 63, 126, 63,
252, 315, 126, 63, 504, 63, 630, 72, 72, 72, 72, 360, 72, 504, 72, 72, 360, 9,
18, 9, 36, 45, 18, 63, 72, 9, 90, 90, 90, 90, 180, 90, 90, 630, 360, 90, 90,
90, 90, 90, 180, 90, 90, 630, 360, 90, 90, 90, 90, 90, 180, 90, 90, 630, 360,
90, 90, 90, 90, 90, 180, 90, 90, 630, 360, 90, 90, 180, 180, 180, 180, 180,
180, 1260, 360, 180, 180, 90, 90, 90, 180, 90, 90, 630, 360, 90, 90, 90, 90,
90, 180, 90, 90, 630, 360, 90, 90, 630, 630, 630, 1260, 630, 630, 630, 2520,
630, 630, 360, 360, 360, 360, 360, 360, 2520, 360, 360, 360, 90, 90, 90, 180,
90, 90, 630, 360, 90, 90, 90, 90, 90, 180, 90, 90, 630, 360, 90, 90, 10, 0, 30,
20, 10, 30, 70, 40, 90, 10, 10, 0, 30, 20, 10, 30, 70, 40, 90, 10, 30, 0, 30,
60, 30, 30, 210, 120, 90, 30, 20, 0, 60, 20, 20, 60, 140, 40, 180, 20, 10, 0,
30, 20, 10, 30, 70, 40, 90, 10, 30, 0, 30, 60, 30, 30, 210, 120, 90, 30, 70, 0,
210, 140, 70, 210, 70, 280, 630, 70, 40, 0, 120, 40, 40, 120, 280, 40, 360, 40,
90, 0, 90, 180, 90, 90, 630, 360, 90, 90, 10, 0, 30, 20, 10, 30, 70, 40, 90,
10, 0, 10, 30, 20, 10, 30, 70, 40, 90, 10, 0, 10, 30, 20, 10, 30, 70, 40, 90,
10, 0, 30, 30, 60, 30, 30, 210, 120, 90, 30, 0, 20, 60, 20, 20, 60, 140, 40,
180, 20, 0, 10, 30, 20, 10, 30, 70, 40, 90, 10, 0, 30, 30, 60, 30, 30, 210,
120, 90, 30, 0, 70, 210, 140, 70, 210, 70, 280, 630, 70, 0, 40, 120, 40, 40,
120, 280, 40, 360, 40, 0, 90, 90, 180, 90, 90, 630, 360, 90, 90, 0, 10, 30, 20,
10, 30, 70, 40, 90, 10, 30, 30, 30, 60, 30, 30, 210, 120, 90, 30, 30, 30, 30, 60,
30, 30, 210, 120, 90, 30, 30, 30, 10, 60, 30, 30, 210, 120, 90, 30, 60, 60, 60,
60, 60, 60, 420, 120, 180, 60, 30, 30, 30, 60, 30, 30, 210, 120, 90, 30, 30,
30, 30, 60, 30, 30, 210, 120, 90, 30, 210, 210, 210, 420, 210, 210, 210, 840,
630, 210, 120, 120, 120, 120, 120, 120, 840, 120, 360, 120, 90, 90, 90, 180,
90, 90, 630, 360, 90, 90, 30, 30, 30, 60, 30, 30, 210, 120, 90, 30, 20, 20, 60,
10, 20, 60, 140, 40, 180, 20, 20, 20, 60, 10, 20, 60, 140, 40, 180, 20, 60, 60,
60, 30, 60, 60, 420, 120, 180, 60, 20, 20, 60, 20, 20, 60, 140, 40, 180, 20,
20, 20, 60, 10, 20, 60, 140, 40, 180, 20, 60, 60, 60, 30, 60, 60, 420, 120,
180, 60, 140, 140, 420, 70, 140, 420, 140, 280, 1260, 140, 40, 40, 120, 40, 40,
120, 280, 40, 360, 40, 180, 180, 180, 90, 180, 180, 1260, 360, 180, 180, 20,
20, 60, 10, 20, 60, 140, 40, 180, 20, 10, 10, 30, 20, 10, 30, 70, 40, 90, 10,
10, 10, 30, 20, 10, 30, 70, 40, 90, 10, 30, 30, 30, 60, 30, 30, 210, 120, 90,
30, 20, 20, 60, 20, 20, 60, 140, 40, 180, 20, 10, 10, 30, 20, 10, 30, 70, 40,
90, 10, 30, 30, 30, 60, 30, 30, 210, 120, 90, 30, 70, 70, 210, 140, 70, 210,
70, 280, 630, 70, 40, 40, 120, 40, 40, 120, 280, 40, 360, 40, 90, 90, 90, 180,
90, 90, 630, 360, 90, 90, 10, 10, 30, 20, 10, 30, 70, 40, 90, 10, 30, 30, 30,
60, 30, 30, 210, 120, 90, 30, 30, 30, 30, 60, 30, 30, 210, 120, 90, 30, 30, 30,
30, 60, 30, 30, 210, 120, 90, 30, 60, 60, 60, 60, 60, 60, 420, 120, 180, 60,
30, 30, 30, 60, 30, 30, 210, 120, 90, 30, 30, 30, 30, 60, 30, 30, 210, 120, 90,
30, 210, 210, 210, 420, 210, 210, 210, 840, 630, 210, 120, 120, 120, 120, 120,
120, 840, 120, 360, 120, 90, 90, 90, 180, 90, 90, 630, 360, 90, 90, 30, 30, 30,
60, 30, 30, 210, 120, 90, 30, 70, 70, 210, 140, 70, 210, 70, 280, 630, 70, 70,
70, 210, 140, 70, 210, 70, 280, 630, 70, 210, 210, 210, 420, 210, 210, 210,
840, 630, 210, 140, 140, 420, 140, 140, 420, 140, 280, 1260, 140, 70, 70, 210,
140, 70, 210, 70, 280, 630, 70, 210, 210, 210, 420, 210, 210, 210, 840, 630,
210, 70, 70, 210, 140, 70, 210, 70, 280, 630, 70, 280, 280, 840, 280, 280, 840,
280, 280, 2520, 280, 630, 630, 630, 1260, 630, 630, 630, 2520, 630, 630, 70,
70, 210, 140, 70, 210, 70, 280, 630, 70, 40, 40, 120, 40, 40, 120, 280, 40,
360, 40, 40, 40, 120, 40, 40, 120, 280, 40, 360, 40, 120, 120, 120, 120, 120,
120, 840, 120, 360, 120, 40, 40, 120, 40, 40, 120, 280, 40, 360, 40, 40, 40,
120, 40, 40, 120, 280, 40, 360, 40, 120, 120, 120, 120, 120, 120, 840, 120,
360, 120, 280, 280, 840, 280, 280, 840, 280, 280, 2520, 280, 40, 40, 120, 40,
40, 120, 280, 40, 360, 40, 360, 360, 360, 360, 360, 360, 2520, 360, 360, 360,
40, 40, 120, 40, 40, 120, 280, 40, 360, 40, 90, 90, 90, 180, 90, 90, 630, 360,
90, 90, 90, 90, 90, 180, 90, 90, 630, 360, 90, 90, 90, 90, 90, 180, 90, 90,
630, 360, 90, 90, 180, 180, 180, 180, 180, 180, 1260, 360, 180, 180, 90, 90,
90, 180, 90, 90, 630, 360, 90, 90, 90, 90, 90, 180, 90, 90, 630, 360, 90, 90,
630, 630, 630, 1260, 630, 630, 630, 2520, 630, 630, 360, 360, 360, 360, 360,
360, 2520, 360, 360, 360, 90, 90, 90, 180, 90, 90, 630, 360, 90, 90, 90, 90,
90, 180, 90, 90, 630, 360, 90, 90, 10, 10, 30, 20, 10, 30, 70, 40, 90, 10, 10,
10, 30, 20, 10, 30, 70, 40, 90, 10, 30, 30, 30, 60, 30, 30, 210, 120, 90, 30,
20, 20, 60, 20, 20, 60, 140, 40, 180, 20, 10, 10, 30, 20, 10, 30, 70, 40, 90,
10, 30, 30, 30, 60, 30, 30, 210, 120, 90, 30, 70, 70, 210, 140, 70, 210, 70,
280, 630, 70, 40, 40, 120, 40, 40, 120, 280, 40, 360, 40, 90, 90, 90, 180, 90,
90, 630, 360, 90, 90, 10, 10, 30, 20, 10, 30, 70, 40, 90, 10, ...
> Integers with
zero distance : 100, 110, 120, 130, 140, 150, 160, 170, 180, 190,
1000, 1001, 1002, 1003, 1004, 1005, 1006, 1007, 1008, 1009, 1011, 1012, 1021,
1022, 1031, 1032, 1041, 1042, 1051, 1052, 1061, 1062, 1071, 1072, 1081, 1082,
1091, 1092, 1100, 1101, 1102, 1103, 1104, 1105, 1106, 1107, 1108, 1109, 1110,
1112, 1120, 1122, 1130, 1132, 1140, 1142, 1150, 1152, 1160, 1162, 1170, 1172,
1180, 1182, 1190, 1192, 1200, 1201, 1202, 1203, 1204, 1205, 1206, 1207, 1208,
1209, 1212, 1222, 1232, 1242, 1252, 1262, 1272, 1282, 1292, 1300, 1301, 1302,
1303, 1304, 1305, 1306, 1307, 1308, 1309, 1312, 1322, 1332, 1342, 1352, 1362,
1372, 1382, 1392, 1400, 1401, 1402, 1403, 1404, 1405, 1406, 1407, 1408, 1409,
1412, 1422, 1432, 1442, 1452, 1462, 1472, 1482, 1492, 1500, 1501, 1502, 1503,
1504, 1505, 1506, 1507, 1508, 1509, 1512, 1522, 1532, 1542, 1552, 1562, 1572,
1582, 1592, 1600, 1601, 1602, 1603, 1604, 1605, 1606, 1607, 1608, 1609, 1612,
1622, 1632, 1642, 1652, 1662, 1672, 1682, 1692, 1700, 1701, 1702, 1703, 1704,
1705, 1706, 1707, 1708, 1709, 1712, 1722, 1732, 1742, 1752, 1762, 1772, 1782,
1792, 1800, 1801, 1802, 1803, 1804, 1805, 1806, 1807, 1808, 1809, 1812, 1822,
1832, 1842, 1852, 1862, 1872, 1882, 1892, 1900, 1901, 1902, 1903, 1904, 1905,
1906, 1907, 1908, 1909, 1912, 1922, 1932, 1942, 1952, 1962, 1972, 1982, 1992,
2000, 2001, 2010, 2011, 2020, 2021, 2030, 2031, 2040, 2041, 2050, 2051, 2060,
2061, 2070, 2071, 2080, 2081, 2090, 2091, 2100, 2101, 2110, 2111, 2120, 2121,
2130, 2131, 2140, 2141, 2150, 2151, 2160, 2161, 2170, 2171, 2180, 2181, 2190,
2191, 2200, 2201, 2210, 2211, 2220, 2221, 2230, 2231, 2240, 2241, 2250, 2251,
2260, 2261, 2270, 2271, 2280, 2281, 2290, 2291, 2300, 2301, 2310, 2311, 2320,
2321, 2330, 2331, 2340, 2341, 2350, 2351, 2360, 2361, 2370, 2371, 2380, 2381,
2390, 2391, 2400, 2401, 2410, 2411, 2420, 2421, 2430, 2431, 2440, 2441, 2450, 2451,
2460, 2461, 2470, 2471, 2480, 2481, 2490, 2491, 2500, 2501, 2510, 2511, 2520,
2521, 2530, 2531, 2540, 2541, 2550, 2551, 2560, 2561, 2570, 2571, 2580, 2581,
2590, 2591, 2600, 2601, 2610, 2611, 2620, 2621, 2630, 2631, 2640, 2641, 2650,
2651, 2660, 2661, 2670, 2671, 2680, 2681, 2690, 2691, 2700, 2701, 2710, 2711,
2720, 2721, 2730, 2731, 2740, 2741, 2750, 2751, 2760, 2761, 2770, 2771, 2780,
2781, 2790, 2791, 2800, 2801, 2810, 2811, 2820, 2821, 2830, 2831, 2840, 2841,
2850, 2851, 2860, 2861, 2870, 2871, 2880, 2881, 2890, 2891, 2900, 2901, 2910,
2911, 2920, 2921, 2930, 2931, 2940, 2941, 2950, 2951, 2960, 2961, 2970, 2971,
2980, 2981, 2990, 2991, 3001, 3011, 3021, 3031, 3041, 3051, 3061, 3071, 3081,
3091, 3100, 3110, 3120, 3130, 3140, 3150, 3160, 3170, 3180, 3190, 4001, 4011,
4021, 4031, 4041, 4051, 4061, 4071, 4081, 4091, 4100, 4110, 4120, 4130, 4140,
4150, 4160, 4170, 4180, 4190, 5001, 5011, 5021, 5031, 5041, 5051, 5061, 5071,
5081, 5091, 5100, 5110, 5120, 5130, 5140, 5150, 5160, 5170, 5180, 5190, 6001,
6011, 6021, 6031, 6041, 6051, 6061, 6071, 6081, 6091, 6100, 6110, 6120, 6130,
6140, 6150, 6160, 6170, 6180, 6190, 7001, 7011, 7021, 7031, 7041, 7051, 7061,
7071, 7081, 7091, 7100, 7110, 7120, 7130, 7140, 7150, 7160, 7170, 7180, 7190,
8001, 8011, 8021, 8031, 8041, 8051, 8061, 8071, 8081, 8091, 8100, 8110, 8120,
8130, 8140, 8150, 8160, 8170, 8180, 8190, 9001, 9011, 9021, 9031, 9041, 9051,
9061, 9071, 9081, 9091, 9100, 9110, 9120, 9130, 9140, 9150, 9160, 9170, 9180,
9190, ...
> Integers which are their own distance (like 18)
> Smallest integer with distance "n"
These might end up being difficult/impossible
to extend. I only have 49 distinct distances < 10000 in my range:
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 18, 20, 21, 24, 28, 30,
35, 36, 40, 42, 45, 56, 60, 63, 70, 72, 84, 90, 105, 120, 126, 140, 168, 180,
210, 252, 280, 315, 360, 420, 504, 630, 840, 1260, 2520.
A points-joined graph:
Many thanks, Hans, for the corrections, results and nice graph!
__________
October 11th
2012, update:
Jean-Paul Davalan, answering me (Eric) on fr.sci.maths
[Eric]:
> Bonjours à tous,
il semblerait que 18 soit le
seul entier inférieur à 10.000 qui soit sa propre distance (de clonage) au sens
expliqué ici. Confirmez-vous ? Lui trouverez-vous des compagnons
au delà de 10.000 ?
[Jean-Paul]:
> Bonjour,
cette distance, c’est pas tout
simplement le ppcm des d+1 (les d sont les chiffres
décimaux) ?
Par exemple, pour 18, le ppcm
de 2=1+1 et de 9=8+1 est 18 ; le ppcm de tous
les d+1 que sont 2, 3, 4, ..., 9, 10 est 8*9*5*7 = 2520, c’est le maximum
possible, et l’ensemble des diviseurs de 2520 est 1, 2, 3, 4, 5, 6, 7, 8, 9,
10, 12, 14, 15, 18, 20, 21, 24, 28, 30, 35, 36, 40, 42, 45, 56, 60, 63, 70, 72,
84, 90, 105, 120, 126, 140, 168, 180, 210, 252, 280, 315, 360, 420, 504, 630,
840, 1260, 2520 ce qui donne, en y ajoutant la valeur 0, la liste des 49
distances que tu as sur ta page, Éric. Il n’y en a pas d’autres, si ?
[Eric, to Hans Havermann,
15 days later]:
> What do you
think, Hans? Are there other distances possible? I guess not, Jean-Paul’s
remark seems sound...
[Hans Havermann]:
> Yes, it’s a
clever observation. I’ll do a 24-hour run past 10000 to convince myself that
there are no counterexamples.
(...)
In the range 10000-99999, I looked for
distances < 12000 other than the 49 we knew. Nothing.
__________
So, Jean-Paul has the answer: there are no
other integers than 18 which are their own distance.
Thanks to all
contributors,
Best,
É.