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 A064384 Primes p such that p divides 0!-1!+2!-3!+...+(-1)^{p-1}(p-1)!. 8
 2, 5, 13, 37, 463 (list; graph; refs; listen; history; text; internal format)
 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, PrimeQ], (Mod[Sum[(-1)^(p - 1)*p!, {p, 2, # - 1}], #] == 0) &] (* Julien Kluge, Feb 13 2016 *) a = 1; a[n_] := a[n] = n*a[n - 1] + 1; Select[Select[Range, PrimeQ], (Mod[a[# - 1], #] == 0) &] (* Julien Kluge, Feb 13 2016 with the sequence approach suggested by Jonathan Sondow *) Select[Prime[Range], 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

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Last modified October 4 03:41 EDT 2023. Contains 365872 sequences. (Running on oeis4.)