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A233263 a(n) = prime(k), where k is such that (Sum_{j=1..k} prime(j)^12) / k is an integer.

%I A233263 #32 Jan 04 2024 18:54:03
%S A233263 2,157,72673,52472909,85790059,88573873,16903607381,4582951241047,
%T A233263 162717490461611,1220077659512857,34871545949176799
%N A233263 a(n) = prime(k), where k is such that (Sum_{j=1..k} prime(j)^12) / k is an integer.
%C A233263 a(11) > 1352363608564489. - _Bruce Garner_, Aug 30 2021
%C A233263 a(12) > 37124508045065437. - _Paul W. Dyson_, Jan 04 2024
%H A233263 OEIS Wiki, <a href="https://oeis.org/wiki/Sums_of_primes_divisibility_sequences">Sums of powers of primes divisibility sequences</a>
%F A233263 a(n) = prime(A131272(n)).
%e A233263 a(2) = 157, because 157 is the 37th prime and the sum of the first 37 primes^12 = 636533120636984811361212036 when divided by 37 equals 17203597855053643550303028 which is an integer.
%p A233263 A233263:=n->if type(add(ithprime(i)^12, i=1..n)/n, integer) then ithprime(n); fi; seq(A233263(n), n=1..100000); # _Wesley Ivan Hurt_, Dec 06 2013
%t A233263 t = {}; sm = 0; Do[sm = sm + Prime[n]^12; If[Mod[sm, n] == 0, AppendTo[t, Prime[n]]], {n, 100000}]; t (* Derived from A217599 *)
%o A233263 (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
%o A233263 (PARI) S=n=0;forprime(p=1,,(S+=p^12)%n++||print1(p",")) \\ _M. F. Hasler_, Dec 01 2013
%Y A233263 Cf. A085450 (smallest m > 1 such that m divides Sum_{k=1..m} prime(k)^n).
%Y A233263 Cf. A007504, A045345, A171399, A128165, A233523, A050247, A050248.
%Y A233263 Cf. A024450, A111441, A217599, A128166, A233862, A217600, A217601.
%K A233263 nonn,more
%O A233263 1,1
%A A233263 _Robert Price_, Dec 06 2013
%E A233263 a(8)-a(9) from _Bruce Garner_, Mar 23 2021
%E A233263 a(10) from _Bruce Garner_, Aug 30 2021
%E A233263 a(11) from _Paul W. Dyson_, Jan 04 2024