A028581 -id:A028581 - cp's OEIS frontend

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

2, 5, 71, 369119, 415074643, 55691042365834801

Offset: 1

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

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

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.

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.

Cf. A024011, A028581, A028582.

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 *)

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

Example corrected by Harvey P. Dale, Feb 09 2013

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

A024011 Numbers k such that the k-th prime divides the sum of the first k primes.

Original entry on oeis.org

1, 3, 20, 31464, 22096548, 1483892396791177

Offset: 1

G. L. Honaker, Jr.

a(6) > pi(10^12) = 37607912018. - Jon E. Schoenfield, Sep 11 2008
a(6) > pi(10^14) = 3204941750802. - Giovanni Resta, Jan 09 2014
a(7) > 6.5*10^15. - Paul W. Dyson, Sep 27 2022

			The third prime, 5, divides 2 + 3 + 5 = 10, so 3 is in the sequence.
2 + 3 + 5 + 7 = 17, which is not divisible by the fourth prime, 7, so 4 is not in the sequence.

Cf. A007506, A028581, A028582, A071089.

s = 0; For[i = 1, i <= 5 * 10^7, i++, s = s + Prime[i]; If[Mod[s, Prime[i + 1]] == 0, Print[i + 1]]]
With[{prs = Prime[Range[221000000]]}, PrimePi /@ Transpose[Select[ Thread[ {Accumulate[prs], prs}], Divisible[#[[1]], #[[2]]] &]][[2]]] (* Harvey P. Dale, Jul 23 2013 *)
nMax = 50000; primeSums = Accumulate[Prime[Range[nMax]]]; Select[Range[nMax], Divisible[primeSums[[#]], Prime[#]] &] (* Alonso del Arte, Nov 11 2019 *)

s=0; t=0; for(w=2,1000000000,if(isprime(w),s=s+w; t=t+1; if(s%w,print(t)),))

a(5) from Kok Seng Chua (chuaks(AT)ihpc.nus.edu.sg), May 14 2000

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

A028582 Dividends associated with A024011.

Original entry on oeis.org

2, 10, 639, 5537154119, 4456255064711219, 40753506500253984833129731848456

Offset: 1

G. L. Honaker, Jr.

Cf. A024011, A007506, A028581.

a(5) from Kok Seng Chua (chuaks(AT)ihpc.nus.edu.sg), May 14 2000

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

cp's OEIS Frontend

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

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A024011 Numbers k such that the k-th prime divides the sum of the first k primes.

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A028582 Dividends associated with A024011.

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