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A233265 Prime(k), where k is such that (1 + Sum_{j=1..k} prime(j)^12) / k is an integer.

%I A233265 #38 Jun 06 2021 06:42:16
%S A233265 2,3,5,7,11,13,17,19,23,29,37,41,43,47,53,61,71,73,89,101,103,107,113,
%T A233265 149,151,167,173,181,197,199,223,239,251,263,281,307,313,317,349,359,
%U A233265 397,409,433,449,463,467,541,569,571,613,643,659,701,733,787,809
%N A233265 Prime(k), where k is such that (1 + Sum_{j=1..k} prime(j)^12) / k is an integer.
%C A233265 a(1171) > 661876608760109. - _Bruce Garner_, Jun 06 2021
%H A233265 Bruce Garner, <a href="/A233265/b233265.txt">Table of n, a(n) for n = 1..1170</a> (terms 1..907 from Robert Price, terms 908..967 from Karl-Heinz Hofmann)
%H A233265 OEIS Wiki, <a href="https://oeis.org/wiki/Sums_of_primes_divisibility_sequences">Sums of powers of primes divisibility sequences</a>
%e A233265 a(4) = 7, because 7 is the 4th prime and (1 + Sum_{i=1..4} prime(i)^12) / 4 = 14085963364/4 = 3521490841 which is an integer.
%p A233265 A233265:=n->if type((1+add(ithprime(i)^12, i=1..n))/n, integer) then ithprime(n); fi; seq(A233265(k),k=1..200); # _Wesley Ivan Hurt_, Dec 06 2013
%t A233265 t = {}; sm = 1; Do[sm = sm + Prime[n]^12; If[Mod[sm, n] == 0, AppendTo[t, Prime[n]]], {n, 100000}]; t (* Derived from A217599 *)
%t A233265 Prime[#]&/@(Flatten[Position[#[[1]]/#[[2]]&/@With[{nn=200},Thread[ {(Rest[ FoldList[ Plus,0,Prime[Range[nn]]^12]])+1,Range[nn]}]],_?IntegerQ]]) (* _Harvey P. Dale_, Nov 19 2018 *)
%o A233265 (PARI) is(n)=if(!isprime(n),return(0)); my(t=primepi(n),s); forprime(p=2,n,s+=Mod(p,t)^12); s==0 \\ _Charles R Greathouse IV_, Nov 30 2013
%Y A233265 Cf. A085450 (smallest m > 1 such that m divides Sum_{k=1..m} prime(k)^n).
%Y A233265 Cf. A007504, A045345, A171399, A128165, A233523, A050247, A050248.
%Y A233265 Cf. A024450, A111441, A217599, A128166, A233862, A217600, A217601.
%K A233265 nonn
%O A233265 1,1
%A A233265 _Robert Price_, Dec 06 2013