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

2, 5, 13, 37, 463

Offset

1,1

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

References

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

Links

Table of n, a(n) for n=1..5.

J. Sondow, The Taylor series for e and the primes 2, 5, 13, 37, 463: a surprising connection

J. 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, Gems in Experimental Mathematics (T. Amdeberhan, L. A. Medina, and V. H. Moll, eds.), Contemporary Mathematics, vol. 517, Amer. Math. Soc., Providence, RI, 2010.

Index entries for sequences related to factorial numbers

Example

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

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 *)

Prog

(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

Crossrefs

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

Sequence in context: A149855 A149856 A320175 * A148302 A149857 A001475

Adjacent sequences: A064381 A064382 A064383 * A064385 A064386 A064387

Keyword

nonn,nice,hard,more

Author

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

Extensions

Edited by Max Alekseyev, Mar 05 2011

Status

approved