A007506 Primes p with property that p divides the sum of all primes <= p. (Formerly M1554)

2, 5, 71, 369119, 415074643, 55691042365834801

Offset

1,1

Comments

a(6) > 29505444491. - Jud McCranie, Jul 08 2000

a(6) > 10^12. - Jon E. Schoenfield, Sep 11 2008

a(6), if it exists, is larger than 10^14. - Giovanni Resta, Jan 09 2014

Also primes p with property that p divides 1 plus the sum of all composites < p. - Vicente Izquierdo Gomez, Aug 05 2014

a(7) > 253814097223614463, - Paul W. Dyson, Sep 27 2022

References

J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 71, p. 25, Ellipses, Paris 2008.

Harry L. Nelson, Prime Sums, J. Rec. Math., 14 (1981), 205-206.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Table of n, a(n) for n=1..6.

H. L. Nelson, Letter to the Editor re: Prime Sums, J. Recreational Mathematics 14.3 (1981-2), 205. (Annotated scanned copy)

Carlos Rivera, Puzzle 18. Some special sums of consecutive primes, The Prime Puzzles and Problems Connection.

Example

2 divides 2;
5 divides 2 + 3 + 5;
71 divides 2 + 3 + 5 + 7 + ... + 61 + 67 + 71; etc.

Mathematica

sumOfPrimes = 0; Do[ sumOfPrimes += p;  If[ Divisible[ sumOfPrimes, p], Print[p]], {p, Prime /@ Range[23000000]}]  (* Jean-François Alcover, Oct 22 2012 *)
Transpose[Module[{nn=23000000, pr}, pr=Prime[Range[nn]]; Select[Thread[ {Accumulate[ pr], pr}], Divisible[#[[1]], #[[2]]]&]]][[2]] (* Harvey P. Dale, Feb 09 2013 *)

Prog

(PARI) s=0; forprime(p=2, 1e9, s+=p; if(s%p==0, print1(p", "))) \\ Charles R Greathouse IV, Jul 22 2013

Crossrefs

Cf. A024011, A028581, A028582.

Sequence in context: A013045 A290864 A309366 * A042693 A328746 A172037

Adjacent sequences: A007503 A007504 A007505 * A007507 A007508 A007509

Keyword

nonn,nice,hard,more

Author

N. J. A. Sloane, Robert G. Wilson v

Extensions

Example corrected by Harvey P. Dale, Feb 09 2013

a(6) from Paul W. Dyson, Apr 16 2022

Status

approved