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%I #45 Feb 07 2024 01:17:13
%S 2,83,241,6599,126551,1544479,4864121,60686737,1966194317,63481708607,
%T 1161468891953,14674403807731,22128836547913,399379081448429,
%U 5410229663058299,248264241666057167
%N Prime(k), where k is such that (Sum_{i=1..k} prime(i)) / k is an integer.
%C Corresponding values of k, Sum_{i=1..k} p_i, and (Sum_{i=1..k} p_i) / k are given in A045345, A050247 and A050248. No other solutions for p_k < 4011201392413.
%C a(13) > 2*10^13. - _Donovan Johnson_, Aug 23 2010
%C a(16) > 5456843462009647. - _Bruce Garner_, Mar 06 2021
%C a(17) > 253814097223614463. - _Paul W. Dyson_, Sep 26 2022
%H Kaisa Matomäki, <a href="http://dx.doi.org/10.1093/qmath/han026">A note on the sum of the first n primes</a>, Quart. J. Math. 61 (2010), pp. 109-115.
%H OEIS Wiki, <a href="https://oeis.org/wiki/Sums_of_primes_divisibility_sequences">Sums of powers of primes divisibility sequences</a>
%F a(n) = A000040(A045345(n)).
%e 83 is the 23rd prime and (Sum_{i=1..23} p_i) / 23 = 874 / 23 = 38 (integer), so 83 is a term.
%t t = {}; sm = 0; Do[sm = sm + Prime[n]; If[Mod[sm, n] == 0, AppendTo[t, Prime[n]]], {n, 100000}]; t (* _T. D. Noe_, Mar 19 2013 *)
%o (PARI) s=0; n=0; forprime(p=2, 1e7, s+=p; if(s%n++==0, print1(p", "))) \\ _Charles R Greathouse IV_, Jun 13 2012
%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.
%Y Cf. A024450, A111441, A217599, A128166, A233862, A217600, A217601.
%K nonn,more
%O 1,1
%A _Jaroslav Krizek_, Dec 07 2009
%E a(6) corrected and a(12) from _Donovan Johnson_, Aug 23 2010
%E a(13) from _Robert Price_, Mar 17 2013
%E a(14)-a(15) from _Bruce Garner_, Mar 06 2021
%E a(16) from _Paul W. Dyson_, Sep 26 2022
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