A007506 - OEIS

%I M1554 #79 Sep 08 2024 12:11:56

%S 2,5,71,369119,415074643,55691042365834801

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

%C a(6) > 29505444491. - _Jud McCranie_, Jul 08 2000

%C a(6) > 10^12. - _Jon E. Schoenfield_, Sep 11 2008

%C a(6), if it exists, is larger than 10^14. - _Giovanni Resta_, Jan 09 2014

%C Also primes p with property that p divides 1 plus the sum of all composites < p.  - _Vicente Izquierdo Gomez_, Aug 05 2014

%C a(7) > 253814097223614463, - _Paul W. Dyson_, Sep 27 2022

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

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

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

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

%H H. L. Nelson, <a href="/A007506/a007506.pdf">Letter to the Editor re: Prime Sums</a>, J. Recreational Mathematics 14.3 (1981-2), 205. (Annotated scanned copy)

%H Carlos Rivera, <a href="http://www.primepuzzles.net/puzzles/puzz_018.htm">Puzzle 18. Some special sums of consecutive primes</a>, The Prime Puzzles and Problems Connection.

%e 2 divides 2;

%e 5 divides 2 + 3 + 5;

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

%t sumOfPrimes = 0; Do[ sumOfPrimes += p;  If[ Divisible[ sumOfPrimes, p], Print[p]], {p, Prime /@ Range[23000000]}]  (* _Jean-François Alcover_, Oct 22 2012 *)

%t Transpose[Module[{nn=23000000,pr},pr=Prime[Range[nn]];Select[Thread[ {Accumulate[ pr], pr}], Divisible[#[[1]],#[[2]]]&]]][[2]] (* _Harvey P. Dale_, Feb 09 2013 *)

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

%Y Cf. A024011, A028581, A028582.

%K nonn,nice,hard,more

%O 1,1

%A _N. J. A. Sloane_, _Robert G. Wilson v_

%E Example corrected by _Harvey P. Dale_, Feb 09 2013

%E a(6) from _Paul W. Dyson_, Apr 16 2022