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%I #56 Sep 03 2022 09:39:42
%S 2,67,157,3217,3637,4201,231947,2790569,30116309,12021325961,
%T 26144296151,1380187561637,6549419699279,735325088697473,
%U 1746583001138813,68725636353488501
%N Prime(n), where n is such that (Sum_{i=1..n} prime(i)^2) / n is an integer.
%C a(16) > 3*10^15 if it exists. - _Anders Kaseorg_, Dec 02 2020
%H OEIS Wiki, <a href="https://oeis.org/wiki/Sums_of_primes_divisibility_sequences">Sums of powers of primes divisibility sequences</a>
%e a(2) = 67, because 67 is the 19th prime and the sum of the first 19 primes^2 = 24966 when divided by 19 equals 1314 which is an integer.
%t t = {}; sm = 0; Do[sm = sm + Prime[n]^2; If[Mod[sm, n] == 0, AppendTo[t, Prime[n]]], {n, 100000}]; t (* _T. D. Noe_, Mar 19 2013 *)
%t k = 1; p = 2; s = 0; lst = {}; While[p < 1000000000, s = s + p^2; If[ Mod[s, k++] == 0, AppendTo[lst, p]]; p = NextPrime@ p]; lst (* _Robert G. Wilson v_, Mar 08 2015 *)
%o (PARI) n=s=0; forprime(p=2,1e9, if((s+=p^2)%n++==0, print1(p", "))) \\ _Charles R Greathouse IV_, Feb 06 2015
%Y Cf. A085450 = smallest m > 1 such that m divides Sum_{k=1..m} prime(k)^n.
%Y Cf. A007504, A045345, A171399, A128165, A233523, A050247, A050248, A024450, A111441, A217599, A128166, A233862, A217600, A217601.
%K nonn,more
%O 1,1
%A _Robert Price_, Mar 19 2013
%E a(13) from _Willem Hengeveld_, Nov 29 2020
%E a(14)-a(15) from _Anders Kaseorg_, Dec 02 2020
%E a(16) from _Paul W. Dyson_, Sep 03 2022
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