Concatenate [integer, first difference]
to form a prime number
Hello SeqFans,
Three thingies:
Let:
a(1) = 1
a = a(n)
b = a(n+1)
b > a
d = b-a
[t,u,...,z] = the concatenation of
the strings 't' and 'u' and ... and 'z'
Now build S(1)
where b is the smallest integer such that [a,d] is
prime
... build
S(2) where b is the smallest integer such that [d,b]
is prime
... build
S(3) where b is the smallest integer such that [a,d,b]
is prime
Are the herunder
starts correct?
S(1) =
1, 2, 5, 8, 11, 14, 23, 26, 29, 32, 49,
...
d = 1 3
3 3 3 9 3 3 3 17
p = 11,23,53,83,113,149,233,263,293,3217,
S(2) =
1, 3, 7, 9, 11, 17, 21, 23, 33, 47, 53, ...
d = 2 4
2 2 6 4
2 10 14 6
p = 23,47,29,211,617,421,223,1033,1447,653,
S(3) =
1, 7,
11, 13, 17,
23, 29, 31,
41, ...
d = 6
4 2 4
6 6 2
10
p = 167,7411,11213,13417,17623,23629,29231,311041,
Best,
É.
__________
[Alois Heinz]:
Dear Eric,
nice new sequences you have, here are the
programs...
S1:= proc(n) option remember;
local
a, d;
if
n=1 then 1
else
a:= S1(n-1);
for
d while not isprime (parse(cat(a,d)))
do
od;
a+d
fi
end:
seq (S1(n), n=1..60);
S1 = 1, 2, 5, 8, 11, 14, 23, 26, 29, 32, 49, 50,
53, 76, 77, 80, 89, 112, 115, 116, 119, 122, 125, 134, 145, 146, 167, 196, 257,
266, 269, 272, 281, 290, 293, 302, 305, 322, 323, 344, 353, 356, 373, 376, 377,
386, 389, 406, 433, 440, 449, 452, 455, 478, 481, 484, 497, 500, 503, 512, ...
__________
S2:= proc(n) option remember;
local
a, d;
if
n=1 then 1
else
a:= S2(n-1);
for
d while not isprime (parse(cat(d,a+d)))
do
od;
a+d
fi
end:
seq (S2(n), n=1..60);
S2 = 1, 3, 7, 9, 11, 17, 21, 23, 33, 47, 53, 57,
61, 63, 67, 69, 71, 77, 83, 87, 91, 93, 101, 111, 113, 131, 141, 143, 1007,
1011, 1013, 1017, 1019, 1023, 1031, 1047, 1051, 1057, 1059, 1061, 1083, 1127,
1131, 1141, 1143, 1157, 1161, 1163, 1169, 1181, 1199, 1203, 1229, 1233, 10027,
10039, 10051, 10053, 10097, 10131, ...
__________
S3:= proc(n) option remember;
local
a, d;
if
n=1 then 1
else
a:= S3(n-1);
for
d while not isprime (parse(cat(a,d,a+d)))
do
od;
a+d
fi
end:
seq (S3(n), n=1..60);
S3 = 1, 7, 11, 13, 17, 23, 29, 31, 41, 43, 49,
59, 71, 79, 91, 1019, 1033, 1073, 1087, 1091, 1093, 1127, 1139, 1163, 1169,
1187, 1193, 1219, 1223, 1237, 1243, 1259, 1301, 1307, 1337, 1339, 1349, 1373,
1403, 1433, 1483, 1489, 1493, 1501, 1547, 1549, 1567, 1577, 1579, 1601, 1631,
1633, 1657, 1661, 1673, 1679, 11683, 11789, 11903, 11911, ...
Best, Alois
__________
... many thanks, Alois, beautiful!
É.
__________
S1 is now http://oeis.org/A173698
S2 is http://oeis.org/A173699