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A064383 Integers n >= 1 such that n divides 0!-1!+2!-3!+4!-...+(-1)^{n-1}(n-1)!.

%I A064383 #24 Jul 02 2023 22:31:23
%S A064383 1,2,4,5,10,13,20,26,37,52,65,74,130,148,185,260,370,463,481,740,926,
%T A064383 962,1852,1924,2315,2405,4630,4810,6019,9260,9620,12038,17131,24076,
%U A064383 30095,34262,60190,68524,85655,120380,171310,222703,342620,445406,890812,1113515
%N A064383 Integers n >= 1 such that n divides 0!-1!+2!-3!+4!-...+(-1)^{n-1}(n-1)!.
%C A064383 If a is in the sequence, then so are all its positive divisors. If a and b are coprime and in the sequence, then so is their product. Hence in extending the sequence, one may as well just look for primes in the sequence (and then check powers of these primes). Heuristically one might expect a very sparse but infinite set of primes in the sequence, but the largest one I know is p=463 and I've searched up to 600000. This sequence was brought to my attention by David Loeffler.
%C A064383 Also, n such that A000522(n)==1 (mod n^2). - _Benoit Cloitre_, Apr 15 2003
%C A064383 The primes in this sequence are the same as the terms > 1 in A124779. - _Jonathan Sondow_, Nov 09 2006
%C A064383 Also, n such that n|A(n-1), where A(0) = 1 and A(k) = k*A(k-1)+1 = A000522(k) for k > 0. - _Jonathan Sondow_, Dec 22 2006
%C A064383 Michael Mossinghoff has calculated that 2, 5, 13, 37, 463 are the only primes in the sequence up to 150 million. - _Jonathan Sondow_, Jun 12 2007
%D A064383 R. K. Guy, Unsolved Problems in Number Theory, 3rd ed., Springer-Verlag, 2004, B43.
%H A064383 J. Sondow, <a href="http://home.earthlink.net/~jsondow/PrimesAndE.pdf">The Taylor series for e and the primes 2, 5, 13, 37, 463: a surprising connection</a>
%H A064383 J. 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>, Gems in Experimental Mathematics (T. Amdeberhan, L. A. Medina, and V. H. Moll, eds.), Contemporary Mathematics, vol. 517, Amer. Math. Soc., Providence, RI, 2010.
%H A064383 <a href="/index/Fa#factorial">Index entries for sequences related to factorial numbers</a>
%F A064383 Up to n=600000, these are just the divisors of 4*5*13*37*463.
%e A064383 4 is in the sequence because 4 divides 0!-1!+2!-3!=1-1+2-6=-4.
%t A064383 s = 0; Do[ s = s + (-1)^(n)(n)!; If[ Mod[ s, n + 1 ] == 0, Print[ n + 1 ] ], {n, 0, 600000} ]
%t A064383 Divisors[4454060] (* From Formula above *) (* _Harvey P. Dale_, Aug 09 2012 *)
%Y A064383 Cf. A000522, A057245, A064384, A124779, A129924.
%K A064383 nonn,nice
%O A064383 1,2
%A A064383 Kevin Buzzard (buzzard(AT)ic.ac.uk), Sep 28 2001
%E A064383 More terms from _Sean A. Irvine_, Jul 02 2023