A064384 Primes p such that p divides 0!-1!+2!-3!+...+(-1)^{p-1}(p-1)!.

2, 5, 13, 37, 463

Offset: 1

Author

Kevin Buzzard (buzzard(AT)ic.ac.uk), Sep 28 2001

Keywords

• nonn
• nice
• hard
• more

Comments

If p is in the sequence then p divides 0!-1!+2!-3!+...+(-1)^N N! for all sufficiently large N. Naive heuristics suggest that the sequence should be infinite but very sparse.

Same as the terms > 1 in A124779. - Jonathan Sondow, Nov 09 2006

A prime p is in the sequence if and only if p|A(p-1), where A(0) = 1 and A(n) = n*A(n-1)+1 = A000522(n). - Jonathan Sondow, Dec 22 2006

Also, a prime p is in this sequence if and only if p divides A061354(p-1). - Alexander Adamchuk, Jun 14 2007

Michael Mossinghoff has calculated that 2, 5, 13, 37, 463 are the only terms up to 150 million. - Jonathan Sondow, Jun 12 2007

Examples

			5 is in the sequence because 5 is prime and it divides 0!-1!+2!-3!+4!=20.
		

References

• R. K. Guy, Unsolved Problems in Theory of Numbers, Springer-Verlag, Third Edition, 2004, B43.

Links

• Jonathan Sondow, The Taylor series for e and the primes 2, 5, 13, 37, 463: a surprising connection
• Jonathan Sondow and K. Schalm, Which partial sums of the Taylor series for e are convergents to e? (and a link to the primes 2, 5, 13, 37, 463), II, arXiv:0709.0671 [math.NT], 2007-2009.
• Index entries for sequences related to factorial numbers

Crossrefs

Cf. A064383, A124779, A000522, A061354, A129924.

Programs

Mathematica

Select[Select[Range[500], PrimeQ], (Mod[Sum[(-1)^(p - 1)*p!, {p, 2, # - 1}], #] == 0) &] (* Julien Kluge, Feb 13 2016 *)
a[0] = 1; a[n_] := a[n] = n*a[n - 1] + 1; Select[Select[Range[500], PrimeQ], (Mod[a[# - 1], #] == 0) &] (* Julien Kluge, Feb 13 2016 with the sequence approach suggested by Jonathan Sondow *)
Select[Prime[Range[500]],Divisible[AlternatingFactorial[#]-1,#]&] (* Harvey P. Dale, Jan 08 2021 *)

PARI

A=1;for(n=1,1000,if(isprime(n),if(Mod(A,n)==0,print(n)));A=n*A+1) \\ Jonathan Sondow, Dec 22 2006

Extensions

Edited by Max Alekseyev, Mar 05 2011