A050248 Integer averages of first k primes for some k.

2, 38, 110, 3066, 60020, 740282, 2340038, 29380602, 957565746, 31043311588, 569424748566, 7207204117608, 10871205353578, 196523412770096, 2665506690112870, 122498079071529726

Offset

1,1

Comments

a(10) and a(11) were found by Giovanni Resta (Nov 15 2004). He states that there are no other terms for primes p < 4011201392413. See link to Prime Puzzles, Puzzle 31 below.

a(13) > (sum of first pi(2*10^13) primes)/pi(2*10^13). - Donovan Johnson, Aug 23 2010

a(16) > 2688482385366706. - Bruce Garner, Mar 06 2021

a(17) > 125237452139872271. - Paul W. Dyson, Sep 26 2022

Links

Table of n, a(n) for n=1..16.

OEIS Wiki, Sums of powers of primes divisibility sequences

Carlos Rivera, Puzzle 31.- The Average Prime number, APN(k) = S(Pk)/k, The Prime Puzzles & Problems Connection.

Eric Weisstein's World of Mathematics, Prime Sums.

Formula

a(n) = A050247(n)/A045345(n).

Example

38 is average of first k = 23 primes; 110 (k = 53); 3066 (k = 853); 60020 (k = 11869).

Mathematica

a=0; lst={}; Do[p=Prime[n]; a+=p; If[a/n==IntegerPart[a/n], AppendTo[lst, a/n]], {n, 10!}]; lst (* Vladimir Joseph Stephan Orlovsky, Aug 05 2009 *)
Module[{nn=10^6, prs}, prs=Prime[Range[nn]]; Select[Table[Mean[Take[prs, n]], {n, nn}], IntegerQ]] (* The program generates the first 7 terms of the sequence. *) (* Harvey P. Dale, Jun 12 2024 *)

Prog

(PARI) s=n=0; forprime(p=2, 1e9, if((s+=p)%n++==0, print1(s/n", "))) \\ Charles R Greathouse IV, Nov 07 2014

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.

Sequence in context: A173026 A294396 A075459 * A337773 A364714 A105645

Adjacent sequences: A050245 A050246 A050247 * A050249 A050250 A050251

Keyword

nonn,more

Author

Eric W. Weisstein

Extensions

Edited by N. J. A. Sloane at the suggestion of David W. Wilson, Jun 23 2007

a(10)-a(11) from Giovanni Resta via Ray Chandler, Jul 19 2010

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

Status

approved