%I #105 Feb 07 2024 01:17:38
%S 1,23,53,853,11869,117267,339615,3600489,96643287,2664167025,
%T 43435512311,501169672991,745288471601,12255356398093,153713440932055,
%U 6361476515268337
%N Numbers k such that k divides sum of first k primes A007504(k).
%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. - _Alexander Adamchuk_, Aug 21 2006
%C a(13) > pi(2*10^13). - _Donovan Johnson_, Aug 23 2010
%C a(15) > 1.42*10^13. - _Giovanni Resta_, Jan 07 2020
%C a(16) > 1.55*10^14. - _Bruce Garner_, Mar 06 2021
%C a(17) > 6.5*10^15. - _Paul W. Dyson_, Sep 26 2022
%C Numbers k such that A090396(k) = 0. - _Felix Fröhlich_, May 05 2021
%H Javier Cilleruelo and Florian Luca, <a href="http://digital.csic.es/bitstream/10261/31070/1/Sum%2520of%2520primes.pdf">On the sum of the first n primes</a>, Q. J. Math. 59:4 (2008), 14 pp.
%H Karl-Heinz Hofmann, <a href="https://www.youtube.com/watch?v=p-Gigy-c1rg">Listening to the terms of A090396</a>, YouTube video, 2021.
%H Kaisa Matomäki, <a href="https://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 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>
%H OEIS wiki, <a href="https://oeis.org/wiki/Sums_of_primes_divisibility_sequences">Sums of powers of primes divisibility sequences</a>.
%F Matomäki proves that a(n) >> n^(24/19). - _Charles R Greathouse IV_, Jun 13 2012
%e 23 is in the sequence because the sum of the first 23 primes is 874 and that's 23 * 38.
%e 53 is in the sequence because the sum of the first 53 primes is 5830 and that's 53 * 110.
%e 83 is not in the sequence because the sum of the first 83 primes is 15968, which leaves a remainder of 32 when divided by 83.
%e The sum of the first a(14) primes is equal to a(14)*196523412770096.
%t s = 0; t = {}; Do[s = s + Prime[n]; If[ Mod[s, n] == 0, AppendTo[t, n]], {n, 1000000}]; t (* _Alexander Adamchuk_, Aug 21 2006 *)
%t nn = 4000000; With[{acpr = Accumulate[Prime[Range[nn]]]}, Select[Range[nn], Divisible[acpr[[#]], #] &]] (* _Harvey P. Dale_, Sep 14 2012 *)
%t Select[Range[100], Mod[Sum[Prime[i], {i, #}], #] == 0 &] (* _Alonso del Arte_, Mar 22 2014 based on _Bill McEachen_'s Wolfram Alpha example *)
%t A007504 = Cases[Import["https://oeis.org/A007504/b007504.txt", "Table"], {_, _}][[All, 2]]; Select[Range[10^5], Divisible[A007504[[# + 1]], #] &] (* _Robert Price_, Mar 13 2020 *)
%o (PARI) s=0;n=0;forprime(p=2,1e7,s+=p;if(s%n++==0,print1(n", "))) \\ _Charles R Greathouse IV_, Jul 15 2011
%o (PARI) isok(n) = (vecsum(primes(n)) % n) == 0; \\ _Michel Marcus_, Nov 26 2020
%o (Python)
%o from itertools import accumulate, count, islice
%o from sympy import prime
%o def A045345_gen(): return (i+1 for i, m in enumerate(accumulate(prime(n) for n in count(1))) if m % (i+1) == 0)
%o A045345_list = list(islice(A045345_gen(),5)) # _Chai Wah Wu_, Feb 23 2022
%Y Cf. A007504, A090396, A171399.
%K nonn,nice,more
%O 1,2
%A _Jud McCranie_
%E More terms from _Alexander Adamchuk_, Aug 21 2006
%E a(12) from _Donovan Johnson_, Aug 23 2010
%E a(13) from _Robert Price_, Mar 17 2013
%E a(14) from _Giovanni Resta_, Jan 07 2020
%E a(15) from _Bruce Garner_, Mar 06 2021
%E a(16) from _Paul W. Dyson_, Sep 26 2022
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