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

1,1

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.

Links

Table of n, a(n) for n=1..45.

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 *)

Prog

(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

Crossrefs

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

Sequence in context: A006275 A042673 A214705 * A007571 A350590 A062621

Adjacent sequences: A252280 A252281 A252282 * A252284 A252285 A252286

Keyword

nonn,base

Author

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

Extensions

More terms from Michel Marcus, Dec 08 2018

Status

approved