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A252283 Intersection of A251964, A252280 and A252281. 5
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 (list; graph; refs; listen; history; text; internal format)
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 A062621 A306748

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

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Last modified July 16 04:26 EDT 2019. Contains 325064 sequences. (Running on oeis4.)