A050248 - OEIS

A050248

Integer averages of first k primes for some k.

106

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

(list; graph; refs; listen; history; text; internal format)

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