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%I #15 Dec 08 2018 11:50:15
%S 2,5,7,241,383,421,523,947,971,1013,1031,1033,1123,1973,2207,2311,
%T 2837,2927,4373,4721,5507,6301,8011,8297,9319,10141,12413,14071,14081,
%U 17957,18311,18353,19163,21013,21401,22501,22901,28211,30103,32027,37699,38083,40507,42797,43321
%N Intersection of A251964, A252280 and A252281.
%C 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
%C be divisible by 11 for k=k_3.
%C Sequence lists primes p for which s(p,k_1)=5, s(p,k_2)=7 and s(p,k_3)=11.
%C 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.
%C For example, for s(p,k_1)=5, s(p,k_2)=7,
%C s(p,k_3)=11, s(p,k_4)=13, s(p,k_5)=17 and
%C s(p,k_6)=19, the known terms of the sequence are 2, 5, 2311, 4721, 43321.
%C A weaker conjecture: {2,5} is the intersection of all such sequences.
%t 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 *)
%o (PARI) s(p,k) = my(s=sumdigits(p^k)); s >> valuation(s, 2);
%o f5(p) = my(k=1); while(s(p,k) % 5, k++); k;
%o isok5(p) = s(p, f5(p)) == 5;
%o f7(p) = my(k=1); while(s(p,k) % 7, k++); k;
%o isok7(p) = s(p, f7(p)) == 7;
%o f11(p) = my(k=1); while(s(p,k) % 11, k++); k;
%o isok11(p) = s(p, f11(p)) == 11;
%o lista(nn) = forprime(p=2, nn, if (isok5(p) && isok7(p) && isok11(p), print1(p, ", "))); \\ _Michel Marcus_, Dec 08 2018
%Y Cf. A221858, A225039, A225093, A251964, A252280, A252281, A252282.
%K nonn,base
%O 1,1
%A _Vladimir Shevelev_ and _Peter J. C. Moses_, Dec 16 2014
%E More terms from _Michel Marcus_, Dec 08 2018
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