A252283 Intersection of A251964, A252280 and A252281.

2, 5, 7, 241, 383, 421, 523, 947, 971, 1013, 1031, 1033, 1123, 1973, 2207, 2311, 2837, 2927, 4373, 4721, 5507, 6301, 8011, 8297, 9319, 10141, 12413, 14071, 14081, 17957, 18311, 18353, 19163, 21013, 21401, 22501, 22901, 28211, 30103, 32027, 37699, 38083, 40507, 42797, 43321

Offset: 1

Author

Vladimir Shevelev and Peter J. C. Moses, Dec 16 2014

Keywords

• nonn
• base

Comments

For a prime p, denote by s(p,k) the odd part of the digital sum of p^k. Let, for the first time, s(p,k) be divisible by 5 for k=k_1, be divisible by 7 for k=k_2 and

be divisible by 11 for k=k_3.

Sequence lists primes p for which s(p,k_1)=5, s(p,k_2)=7 and s(p,k_3)=11.

Consider also sequence which lists primes p with s(p,k_1)=5, s(p,k_2)=7, s(p,k_3)=11 and s(p,k_4)=13; sequence which lists primes p with s(p,k_1)=5, s(p,k_2)=7, s(p,k_3)=11, s(p,k_4)=13 and s(p,k_5)=17; etc. Then it seems that we will be eventually left with 2 and 5.

For example, for s(p,k_1)=5, s(p,k_2)=7,

s(p,k_3)=11, s(p,k_4)=13, s(p,k_5)=17 and

s(p,k_6)=19, the known terms of the sequence are 2, 5, 2311, 4721, 43321.

A weaker conjecture: {2,5} is the intersection of all such sequences.

Crossrefs

Cf. A221858, A225039, A225093, A251964, A252280, A252281, A252282.

Programs

Mathematica

s[p_, k_] := Module[{s = Total[IntegerDigits[p^k]]}, s/2^IntegerExponent[s, 2]]; f[p_, q_] := Module[{k = 1}, While[! Divisible[s[p, k], q], k++]; k]; okQ[p_, q_] := s[p, f[p, q]] == q; Select[Range[2400],  PrimeQ[#] && okQ[#, 5] && okQ[#, 7] && okQ[#, 11] &] (* Amiram Eldar, Dec 08 2018 *)

PARI

s(p,k) = my(s=sumdigits(p^k)); s >> valuation(s, 2);
f5(p) = my(k=1); while(s(p,k) % 5, k++); k;
isok5(p) = s(p, f5(p)) == 5;
f7(p) = my(k=1); while(s(p,k) % 7, k++); k;
isok7(p) = s(p, f7(p)) == 7;
f11(p) = my(k=1); while(s(p,k) % 11, k++); k;
isok11(p) = s(p, f11(p)) == 11;
lista(nn) = forprime(p=2, nn, if (isok5(p) && isok7(p) && isok11(p), print1(p, ", "))); \\ Michel Marcus, Dec 08 2018

Extensions

More terms from Michel Marcus, Dec 08 2018