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A045345 Numbers n such that n divides sum of first n primes A007504(n). 111
1, 23, 53, 853, 11869, 117267, 339615, 3600489, 96643287, 2664167025, 43435512311, 501169672991, 745288471601 (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

LINKS

Table of n, a(n) for n=1..13.

Javier Cilleruelo and Florian Luca, On the sum of the first n primes, Q. J. Math. 59:4 (2008), 14 pp.

Kaisa Matomäki, A note on the sum of the first n primes, Quart. J. Math. 61 (2010), pp. 109-115.

The Prime Puzzles & Problems Connection: Puzzle 31.- The Average Prime number, APN(k) = S(Pk)/k.

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.

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 *)

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

CROSSREFS

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

STATUS

approved

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Last modified December 22 02:03 EST 2014. Contains 252326 sequences.