

A045345


Numbers k such that k divides sum of first k primes A007504(k).


124



1, 23, 53, 853, 11869, 117267, 339615, 3600489, 96643287, 2664167025, 43435512311, 501169672991, 745288471601, 12255356398093, 153713440932055, 6361476515268337
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OFFSET

1,2


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.  Alexander Adamchuk, Aug 21 2006
a(13) > pi(2*10^13).  Donovan Johnson, Aug 23 2010
a(15) > 1.42*10^13.  Giovanni Resta, Jan 07 2020
a(16) > 1.55*10^14.  Bruce Garner, Mar 06 2021
a(17) > 6.5*10^15.  Paul W. Dyson, Sep 26 2022
Numbers k such that A090396(k) = 0.  Felix Fröhlich, May 05 2021


LINKS

Table of n, a(n) for n=1..16.
Javier Cilleruelo and Florian Luca, On the sum of the first n primes, Q. J. Math. 59:4 (2008), 14 pp.
KarlHeinz 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. 109115.
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.


FORMULA

Matomäki proves that a(n) >> n^(24/19).  Charles R Greathouse IV, Jun 13 2012


EXAMPLE

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.


MAPLE

with(numtheory); ListA045345:=proc(q) local k, n;
for n from 1 to q do if add(ithprime(k), k=1..n) mod n=0 then print(n);
fi; od; end: ListA045345(10^12); # Paolo P. Lava, Jun 27 2013


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 *)


PROG

(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


CROSSREFS

Cf. A007504, A090396, A171399.
Sequence in context: A078854 A078959 A238854 * A133986 A103006 A053236
Adjacent sequences: A045342 A045343 A045344 * A045346 A045347 A045348


KEYWORD

nonn,nice,more


AUTHOR

Jud McCranie


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


STATUS

approved



