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%I #20 Jul 07 2017 13:15:01
%S 3,2,1,3,7,4,9,1,11,6,1,7,3,8,17,9,19,2,21,11,23,12,5,1,27,14,29,3,31,
%T 16,33,17,7,18,1,19,3,4,41,21,43,22,9,23,47,24,49,5,51,2,53,27,11,28,
%U 57,29,59,6,61,31,63,32,1,33,67,34,69,7,71,36,73,1,15,38,77,3,79,8,81,41
%N a(n) = (n+3)/gcd(A(n), A(n+2)) where A(n) = A000522(n) = Sum_{k=0..n} n!/k!.
%C a(n) is an integer since A(n+2) = (n+2)(n+1)*A(n) + n+3.
%H Antti Karttunen, <a href="/A124782/b124782.txt">Table of n, a(n) for n = 0..4096</a>
%H J. Sondow, <a href="http://www.jstor.org/stable/27642006">A geometric proof that e is irrational and a new measure of its irrationality</a>, Amer. Math. Monthly 113 (2006) 637-641.
%H J. Sondow, <a href="http://arxiv.org/abs/0704.1282">A geometric proof that e is irrational and a new measure of its irrationality</a>, [math.HO], 2007-2010.
%H 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 <a href="/index/Fa#factorial">Index entries for sequences related to factorial numbers</a>
%F a(n) = (n+3)/A124780(n) = (n+3)/gcd(A000522(n), A000522(n+2)).
%e a(3) = (3+3)/gcd(A(3), A(5)) = 6/gcd(16, 326) = 6/2 = 3.
%t (A[n_] := Sum[n!/k!, {k,0,n}]; Table[(n+3)/GCD[A[n],A[n+2]], {n,0,80}])
%o (PARI)
%o A000522(n) = sum(k=0, n, binomial(n, k)*k!); \\ This function from _Joerg Arndt_, Dec 14 2014
%o A124780(n) = gcd(A000522(n),A000522(n+2));
%o A124782(n) = ((n+3)/A124780(n)); \\ _Antti Karttunen_, Jul 07 2017
%Y Cf. A000522, A093101, A123899, A123900, A123901, A124779, A124780, A124781.
%K nonn
%O 0,1
%A _Jonathan Sondow_, Nov 07 2006
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