This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A064384 #37 Oct 29 2023 10:50:34
%S A064384 2,5,13,37,463
%N A064384 Primes p such that p divides 0!-1!+2!-3!+...+(-1)^{p-1}(p-1)!.
%C A064384 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.
%C A064384 Same as the terms > 1 in A124779. - _Jonathan Sondow_, Nov 09 2006
%C A064384 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
%C A064384 Also, a prime p is in this sequence if and only if p divides A061354(p-1). - _Alexander Adamchuk_, Jun 14 2007
%C A064384 Michael Mossinghoff has calculated that 2, 5, 13, 37, 463 are the only terms up to 150 million. - _Jonathan Sondow_, Jun 12 2007
%D A064384 R. K. Guy, Unsolved Problems in Theory of Numbers, Springer-Verlag, Third Edition, 2004, B43.
%H A064384 Jonathan Sondow, <a href="https://web.archive.org/web/20200215083228/https://home.earthlink.net/~jsondow/PrimesAndE.pdf">The Taylor series for e and the primes 2, 5, 13, 37, 463: a surprising connection</a>
%H A064384 Jonathan Sondow and K. Schalm, <a href="http://arxiv.org/abs/0709.0671">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</a>, arXiv:0709.0671 [math.NT], 2007-2009.
%H A064384 <a href="/index/Fa#factorial">Index entries for sequences related to factorial numbers</a>
%e A064384 5 is in the sequence because 5 is prime and it divides 0!-1!+2!-3!+4!=20.
%t A064384 Select[Select[Range[500], PrimeQ], (Mod[Sum[(-1)^(p - 1)*p!, {p, 2, # - 1}], #] == 0) &] (* _Julien Kluge_, Feb 13 2016 *)
%t A064384 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_ *)
%t A064384 Select[Prime[Range[500]],Divisible[AlternatingFactorial[#]-1,#]&] (* _Harvey P. Dale_, Jan 08 2021 *)
%o A064384 (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
%Y A064384 Cf. A064383, A124779, A000522, A061354, A129924.
%K A064384 nonn,nice,hard,more
%O A064384 1,1
%A A064384 Kevin Buzzard (buzzard(AT)ic.ac.uk), Sep 28 2001
%E A064384 Edited by _Max Alekseyev_, Mar 05 2011