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A233557 Prime(k), where k is such that (1 + Sum_{i=1..k} prime(i)^17) / k is an integer.

%I A233557 #37 Jun 02 2021 05:47:22
%S A233557 2,3,7,13,29,37,641,853,2143,18059,26417,34283,48539,122597,146539,
%T A233557 254831,8304757,19534651,26528699,32820527,47825363,82199141,
%U A233557 124088207,312168289,409464961,464174839,1167927947,1393486043,1725361103,1879982849,4346448019,7331901341,7451088943,27036461983,39662532977,113692593373,449281234057
%N A233557 Prime(k), where k is such that (1 + Sum_{i=1..k} prime(i)^17) / k is an integer.
%C A233557 a(45) > 491952295618219. - _Bruce Garner_, Jun 02 2021
%H A233557 Bruce Garner, <a href="/A233557/b233557.txt">Table of n, a(n) for n = 1..44</a> (first 37 terms from Robert Price, terms 38..39 from Karl-Heinz Hofmann)
%H A233557 OEIS Wiki, <a href="https://oeis.org/wiki/Sums_of_primes_divisibility_sequences">Sums of powers of primes divisibility sequences</a>
%e A233557 13 is a term because 13 is the 6th prime and the sum of the first 6 primes^17+1 = 9156096341463343272 when divided by 6 equals 1526016056910557212 which is an integer.
%t A233557 t = {}; sm = 1; Do[sm = sm + Prime[n]^17; If[Mod[sm, n] == 0, AppendTo[t, Prime[n]]], {n, 100000}]; t (* Derived from A217599 *)
%t A233557 With[{nn=175*10^8},Prime[#]&/@Select[Thread[{Range[nn],Accumulate[ Prime[ Range[nn]]^17]}],Divisible[#[[2]]+1,#[[1]]]&][[All,1]]] (* The program will take a long time to run *) (* _Harvey P. Dale_, Apr 13 2018 *)
%o A233557 (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
%Y A233557 Cf. A085450 (smallest m > 1 such that m divides Sum_{k=1..m} prime(k)^n).
%Y A233557 Cf. A007504, A045345, A171399, A128165, A233523, A050247, A050248.
%Y A233557 Cf. A024450, A111441, A217599, A128166, A233862, A217600, A217601.
%K A233557 nonn
%O A233557 1,1
%A A233557 _Robert Price_, Dec 12 2013