1,1

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. I have searched up to 600000 and have not found another prime with this property.

Same as the terms > 1 in A124779. - Jonathan Sondow (jsondow(AT)alumni.princeton.edu), 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 (jsondow(AT)alumni.princeton.edu), Dec 22 2006

It appears that also a(n) = primes p such that p divides A061354(p-1), where A061354(n) = Numerator of Sum_{k=0..n} 1/k!. - Alexander Adamchuk (alex(AT)kolmogorov.com), Jun 14 2007

Michael Mossinghoff has calculated that 2, 5, 13, 37, 463 are the only terms up to 150 million. - Jonathan Sondow (jsondow(AT)alumni.princeton.edu), Jun 12 2007

For proofs of Adamchuk's and my Comments, see the link "The Taylor series for e ...". - Jonathan Sondow (jsondow(AT)alumni.princeton.edu), Jun 18 2007

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

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

Index entries for sequences related to factorial numbers

J. Sondow, Which Partial Sums of the Taylor Series for e Are Convergents to e? (and a Link to the Primes $2, 5, 13, 37, 463, ...$) with an Appendix "Periodic Behaviour of Some Recurrence Sequences Related to $e$, Modulo Powers of 2" by Kyle Schalm

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

(PARI) gp > A=1; for(n=1, 1000, if(isprime(n), if(Mod(A, n)==0, print(n))); A=n*A+1) - Jonathan Sondow (jsondow(AT)alumni.princeton.edu), Dec 22 2006

Cf. A064383, A124779.

Cf. A000522.

Cf. A061354 = Numerator of Sum_{k=0..n} 1/k!.

Cf. A129924.

Sequence in context: A003080 A053732 A119495 this_sequence A001475 A000800 A006823

Adjacent sequences: A064381 A064382 A064383 this_sequence A064385 A064386 A064387

nonn,nice

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