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A045345 Numbers k such that k divides sum of first k primes A007504(k). 121
1, 23, 53, 853, 11869, 117267, 339615, 3600489, 96643287, 2664167025, 43435512311, 501169672991, 745288471601, 12255356398093, 153713440932055 (list; graph; refs; listen; history; text; internal format)
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

Numbers k such that A090396(k) = 0. - Felix Fröhlich, May 05 2021

LINKS

Table of n, a(n) for n=1..15.

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.

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

CROSSREFS

Cf. A007504, A090396.

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

STATUS

approved

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Last modified December 7 13:08 EST 2021. Contains 349581 sequences. (Running on oeis4.)