A007506 Primes p with property that p divides the sum of all primes <= p.

2, 5, 71, 369119, 415074643, 55691042365834801

Offset: 1

Author

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

Keywords

• nonn
• nice
• hard
• more

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

Examples

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

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).
• David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See p. 129.

Links

• 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.

Crossrefs

Cf. A024011, A028581, A028582.

Programs

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 *)

PARI

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

Extensions

Example corrected by Harvey P. Dale, Feb 09 2013

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