A171399 Prime(k), where k is such that (Sum_{i=1..k} prime(i)) / k is an integer.

2, 83, 241, 6599, 126551, 1544479, 4864121, 60686737, 1966194317, 63481708607, 1161468891953, 14674403807731, 22128836547913, 399379081448429, 5410229663058299, 248264241666057167

Offset: 1

Author

Jaroslav Krizek, Dec 07 2009

Keywords

• nonn
• more

Comments

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.

a(13) > 2*10^13. - Donovan Johnson, Aug 23 2010

a(16) > 5456843462009647. - Bruce Garner, Mar 06 2021

a(17) > 253814097223614463. - Paul W. Dyson, Sep 26 2022

Examples

			83 is the 23rd prime and (Sum_{i=1..23} p_i) / 23 = 874 / 23 = 38 (integer), so 83 is a term.
		

Links

• Kaisa Matomäki, A note on the sum of the first n primes, Quart. J. Math. 61 (2010), pp. 109-115.
• OEIS Wiki, Sums of powers of primes divisibility sequences

Crossrefs

Cf. A085450 (smallest m > 1 such that m divides Sum_{k=1..m} prime(k)^n).

Cf. A007504, A045345, A171399, A128165, A233523, A050247, A050248.

Cf. A024450, A111441, A217599, A128166, A233862, A217600, A217601.

Programs

Mathematica

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 *)

PARI

s=0; n=0; forprime(p=2, 1e7, s+=p; if(s%n++==0, print1(p", "))) \\ Charles R Greathouse IV, Jun 13 2012

Formula

a(n) = A000040(A045345(n)).

Extensions

a(6) corrected and a(12) from Donovan Johnson, Aug 23 2010

a(13) from Robert Price, Mar 17 2013

a(14)-a(15) from Bruce Garner, Mar 06 2021

a(16) from Paul W. Dyson, Sep 26 2022