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%I #24 Jul 13 2017 21:11:34
%S 1,1,1,2,1,2,1,20,1,10,1,8,5,2,5,4,1,130,1,4000,1,2,5,52,5,494,1,40,1,
%T 10,13,4,25,38,5,16,13,230,13,20,1,46,5,104,475,62,1,20,1,130,31,832,
%U 2755,74,5,4,13,50,1,40,23,2,2795,76,34385,2,1,80,1,650,1,2812,5,74,5
%N Cancellation factor in reducing Sum_{k=0...n} 1/k! to lowest terms.
%C Same as n!/A061355(n) and (1+n+n(n-1)+n(n-1)(n-2)+...+n!)/A061354(n).
%C a(n) is relatively prime to n.
%C gcd(a(n),a(n+1)) = 1.
%H Antti Karttunen, <a href="/A093101/b093101.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>, arXiv:0704.1282 [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.
%F a(n) = gcd(n!, 1+n+n(n-1)+n(n-1)(n-2)+...+n!).
%F a(n) = gcd(n!, A(n)) where A(0) = 1, A(n) = n*A(n-1)+1.
%e E.g. 1/0!+1/1!+1/2!+1/3!=16/6=(2*8)/(2*3) so a(3)=2.
%t f[n_] := n! / Denominator[ Sum[1/k!, {k, 0, n}]]; Table[ f[n], {n, 0, 74}] (* _Robert G. Wilson v_ *)
%t (* Second program: *)
%t A[n_] := If[n==0,1,n*A[n-1]+1]; Table[GCD[A[n],n! ], {n, 0, 74}]
%o (PARI)
%o A000522(n) = sum(k=0, n, binomial(n, k)*k!); \\ This function from _Joerg Arndt_, Dec 14 2014
%o A093101(n) = gcd(n!,A000522(n)); \\ _Antti Karttunen_, Jul 12 2017
%Y Cf. A093647, A093651.
%Y (n+1)!/(a(n)*a(n+1)) = A123899(n).
%Y (n+3)!/(a(n)*a(n+1)*a(n+2)) = A123900(n).
%Y (n+3)/GCD(a(n), a(n+2)) = A123901(n).
%Y Cf. also A000522, A061354, A061355.
%K nonn
%O 0,4
%A _Jonathan Sondow_, May 10 2004, Oct 18 2006
%E More terms from _Robert G. Wilson v_, May 14 2004
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