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%I #20 Jun 04 2019 09:32:40
%S 0,1,2,4,6,8,10,16,18,20,26,28,40,46,48,58,66,68,70,80,86,96,98,118,
%T 126,130,136,146,150,170,176,178,180,188,190,206,208,210,216,230,260,
%U 266,268,278,286,288,300,306,308,326,328,338,346,358,366,370,378,380,388
%N Numbers n such that denominator of Sum_{k=0 to n} 1/k! is n!.
%C a(n) is even for n > 1, as Sum_{k=0 to n} 1/k! reduces to lower terms when n > 1 is odd.
%H J. Sondow, <a href="https://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="https://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.
%H <a href="/index/Fa#factorial">Index entries for sequences related to factorial numbers</a>
%F a(n) = 2*A102471(n-1) for n > 1.
%e 1/0! + 1/1! + 1/2! + 1/3! +1/4! = 65/24 and 24 = 4!, so 4 is a member. But 1/0! + 1/1! + 1/2! + 1/3! = 8/3 and 3 < 3!, so 3 is not a member.
%t fQ[n_] := (Denominator[Sum[1/k!, {k, 0, n}]] == n!); Select[ Range[0, 389], fQ[ # ] &] (* _Robert G. Wilson v_, Jan 15 2005 *)
%Y For n > 0, n is a member <=> A093101(n) = 1 <=> A061355(n) = n! <=> A061355(n) = A002034(A061355(n))! <=> A061354(n) = 1+n+n(n-1)+n(n-1)(n-2)+...+n!. See also A102471.
%K nonn
%O 1,3
%A _Jonathan Sondow_, Jan 14 2005
%E More terms from _Robert G. Wilson v_, Jan 15 2005
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