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A233555 Prime(m), where m is such that (Sum_{i=1..m} prime(i)^17) / m is an integer.

%I A233555 #46 Sep 15 2023 21:30:36
%S A233555 2,5724469,10534369,16784723,33330911,189781037,8418091991,
%T A233555 58605633953,109388266843,448366797199,1056238372873,24603683667221,
%U A233555 86982253895059,100316149840769,164029709175817,542295448805641,685217940914237,1701962315686097,23064173255594491
%N A233555 Prime(m), where m is such that (Sum_{i=1..m} prime(i)^17) / m is an integer.
%C A233555 a(18) > 1005368767096627. - _Bruce Garner_, Aug 30 2021
%C A233555 a(19) > 1701962315686097. - _Bruce Garner_, Jan 07 2022
%H A233555 OEIS Wiki, <a href="https://oeis.org/wiki/Sums_of_primes_divisibility_sequences">Sums of powers of primes divisibility sequences</a>
%F A233555 a(n) = prime(A131277(n)).
%e A233555 a(1) = 2, because 2 is the 1st prime and the sum of the first 1 primes^17 = 131072 when divided by 1 equals 131072 which is an integer.
%t A233555 t = {}; sm = 0; Do[sm = sm + Prime[n]^17; If[Mod[sm, n] == 0, AppendTo[t, Prime[n]]], {n, 100000}]; t (* Derived from A217599 *)
%o A233555 (PARI) is(n)=if(!isprime(n),return(0)); my(t=primepi(n),s); forprime(p=2,n,s+=Mod(p,t)^17); s==0 \\ _Charles R Greathouse IV_, Nov 30 2013
%o A233555 (PARI) S=n=0;forprime(p=1,,(S+=p^17)%n++||print1(p",")) \\ _M. F. Hasler_, Dec 01 2013
%Y A233555 Cf. A085450 (smallest m > 1 that divide Sum_{k=1..m} prime(k)^n).
%Y A233555 Cf. A000040, A007504, A045345, A171399, A128165, A233523, A050247, A050248.
%Y A233555 Cf. A024450, A111441, A217599, A128166, A233862, A217600, A217601.
%K A233555 nonn
%O A233555 1,1
%A A233555 _Robert Price_, Dec 12 2013
%E A233555 a(12) from _Bruce Garner_, Mar 02 2021
%E A233555 a(13) from _Bruce Garner_, Mar 17 2021
%E A233555 a(14) from _Bruce Garner_, Mar 30 2021
%E A233555 a(15) from _Bruce Garner_, Apr 14 2021
%E A233555 a(16) from _Bruce Garner_, Jun 30 2021
%E A233555 a(17) from _Bruce Garner_, Aug 30 2021
%E A233555 a(18) from _Bruce Garner_, Jan 07 2022
%E A233555 a(19) from _Paul W. Dyson_, Sep 15 2023