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%I #49 Sep 27 2022 09:01:11
%S 2,874,5830,2615298,712377380,86810649294,794712005370,
%T 105784534314378,92542301212047102,82704567079549985700,
%U 24733255676526572596026,3612032130800990065125528,8102184022428756416738378
%N a(n) is the sum of the first A045345(n) primes.
%C 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.
%C a(13) > 6640510710493148698166596 (sum of first pi(2*10^13) primes). - _Donovan Johnson_, Aug 23 2010
%C a(16) > 416714769731839517991408161209 (sum of first pi(1.55*10^14) primes). - _Bruce Garner_, Mar 06 2021
%C a(17) > 814043439429001245436559390420866 (sum of first 6500000004150767 primes). - _Paul W. Dyson_, Sep 27 2022
%H Paul W. Dyson, <a href="/A050247/b050247.txt">Table of n, a(n) for n = 1..16</a> (terms 1..15 from Bruce Garner).
%H OEIS Wiki, <a href="https://oeis.org/wiki/Sums_of_primes_divisibility_sequences">Sums of powers of primes divisibility sequences</a>
%H Carlos Rivera, <a href="http://www.primepuzzles.net/puzzles/puzz_031.htm">Puzzle 31.- The Average Prime number, APN(k) = S(Pk)/k</a>, The Prime Puzzles & Problems Connection.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PrimeSums.html">Prime Sums</a>
%F a(n) = Sum_{i=1..A045345(n)} A000040(i). - _R. J. Mathar_, Jan 26 2008
%t A007504 = Cases[Import["https://oeis.org/A007504/b007504.txt", "Table"], {_, _}][[All, 2]];
%t A045345 = Cases[Import["https://oeis.org/A045345/b045345.txt", "Table"], {_, _}][[All, 2]];
%t Table[A007504[[A045345[[n]] + 1]], {n, 1, 5}](* _Robert Price_, Mar 13 2020 *)
%o (PARI) s=n=0; forprime(p=2, 1e9, if((s+=p)%n++==0, print1(s", "))) \\ _Charles R Greathouse IV_, Nov 07 2014
%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, A024450, A111441, A217599, A128166, A233862, A217600, A217601.
%K nonn,more
%O 1,1
%A _Eric W. Weisstein_
%E a(10)-a(11) from _Giovanni Resta_ submitted by _Ray Chandler_, Jul 19 2010
%E a(12) from _Donovan Johnson_, Aug 23 2010
%E a(13) from _Robert Price_, Mar 17 2013
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