A252283 - OEIS

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

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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