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A233767 Prime(n), where n is such that (Sum_{i=1..n} prime(i)^19) / n is an integer.

%I A233767 #50 Jan 01 2025 03:27:11
%S A233767 2,97,3203,5059,6469,8081,35051,39719,42209,109049,154591,523297,
%T A233767 6621827,20059771,258196441,731584957,1427109029,1899496631,
%U A233767 8428550519,50790885203,7475902096387,22626378502139,38855796912367,162082298018497,589085299527401,4271778258271487
%N A233767 Prime(n), where n is such that (Sum_{i=1..n} prime(i)^19) / n is an integer.
%C A233767 a(26) > 661876608760109. - _Bruce Garner_, Jun 30 2021
%C A233767 a(27) > 18205684894350047. - _Paul W. Dyson_, Dec 31 2024
%H A233767 OEIS Wiki, <a href="https://oeis.org/wiki/Sums_of_primes_divisibility_sequences">Sums of powers of primes divisibility sequences</a>
%F A233767 a(n) = prime(A131279(n)).
%e A233767 97 is a term, because 97 is the 25th prime and the sum of the first 25 primes^19 = 71486619210134792705255313675343157050 when divided by 25 equals 2859464768405391708210212547013726282 which is an integer.
%t A233767 t = {}; sm = 0; Do[sm = sm + Prime[n]^19; If[Mod[sm, n] == 0, AppendTo[t, Prime[n]]], {n, 100000}]; t (* Derived from A217599 *)
%o A233767 (PARI) is(n)=if(!isprime(n),return(0)); my(t=primepi(n),s); forprime(p=2,n,s+=Mod(p,t)^19); s==0 \\ _Charles R Greathouse IV_, Nov 30 2013
%o A233767 (PARI) my(S=n=0);forprime(p=1,,(S+=p^19)%n++||print1(p", ")) \\ _M. F. Hasler_, Dec 01 2013
%Y A233767 Cf. A085450 (smallest m > 1 such that m divides Sum_{k=1..m} prime(k)^n).
%Y A233767 Cf. A007504, A045345, A171399, A128165, A233523, A050247, A050248.
%Y A233767 Cf. A024450, A111441, A217599, A128166, A233862, A217600, A217601.
%K A233767 nonn,less
%O A233767 1,1
%A A233767 _Robert Price_, Dec 15 2013
%E A233767 a(21) from _Karl-Heinz Hofmann_, Feb 24 2021
%E A233767 a(22) from _Bruce Garner_, Mar 01 2021
%E A233767 a(23) from _Bruce Garner_, Mar 08 2021
%E A233767 a(24) from _Bruce Garner_, Apr 14 2021
%E A233767 a(25) from _Bruce Garner_, Jun 30 2021
%E A233767 a(26) from _Paul W. Dyson_, Jun 27 2023