A045345 Numbers k such that k divides sum of first k primes A007504(k).
1, 23, 53, 853, 11869, 117267, 339615, 3600489, 96643287, 2664167025, 43435512311, 501169672991, 745288471601, 12255356398093, 153713440932055, 6361476515268337
Offset: 1
Examples
23 is in the sequence because the sum of the first 23 primes is 874 and that's 23 * 38. 53 is in the sequence because the sum of the first 53 primes is 5830 and that's 53 * 110. 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. The sum of the first a(14) primes is equal to a(14)*196523412770096.
Links
- Javier Cilleruelo and Florian Luca, On the sum of the first n primes, Q. J. Math. 59:4 (2008), 14 pp.
- Karl-Heinz Hofmann, Listening to the terms of A090396, YouTube video, 2021.
- Kaisa Matomäki, A note on the sum of the first n primes, Quart. J. Math. 61 (2010), pp. 109-115.
- 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
- OEIS wiki, Sums of powers of primes divisibility sequences.
Programs
-
Mathematica
s = 0; t = {}; Do[s = s + Prime[n]; If[ Mod[s, n] == 0, AppendTo[t, n]], {n, 1000000}]; t (* Alexander Adamchuk, Aug 21 2006 *) nn = 4000000; With[{acpr = Accumulate[Prime[Range[nn]]]}, Select[Range[nn], Divisible[acpr[[#]], #] &]] (* Harvey P. Dale, Sep 14 2012 *) Select[Range[100], Mod[Sum[Prime[i], {i, #}], #] == 0 &] (* Alonso del Arte, Mar 22 2014 based on Bill McEachen's Wolfram Alpha example *) A007504 = Cases[Import["https://oeis.org/A007504/b007504.txt", "Table"], {, }][[All, 2]]; Select[Range[10^5], Divisible[A007504[[# + 1]], #] &] (* Robert Price, Mar 13 2020 *) -
PARI
s=0;n=0;forprime(p=2,1e7,s+=p;if(s%n++==0,print1(n", "))) \\ Charles R Greathouse IV, Jul 15 2011
-
PARI
isok(n) = (vecsum(primes(n)) % n) == 0; \\ Michel Marcus, Nov 26 2020
-
Python
from itertools import accumulate, count, islice from sympy import prime def A045345_gen(): return (i+1 for i, m in enumerate(accumulate(prime(n) for n in count(1))) if m % (i+1) == 0) A045345_list = list(islice(A045345_gen(),5)) # Chai Wah Wu, Feb 23 2022
Formula
Matomäki proves that a(n) >> n^(24/19). - Charles R Greathouse IV, Jun 13 2012
Extensions
More terms from Alexander Adamchuk, Aug 21 2006
a(12) from Donovan Johnson, Aug 23 2010
a(13) from Robert Price, Mar 17 2013
a(14) from Giovanni Resta, Jan 07 2020
a(15) from Bruce Garner, Mar 06 2021
a(16) from Paul W. Dyson, Sep 26 2022
Comments