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%I #20 Jun 04 2019 10:27:37
%S 1,2,6,10,30,32,42,46,56,62,70,80,82,96,120,122,136,150,160,162,170,
%T 172,176,186,192,196,200,210,222,230,236,252,262,266,276,290,292,300,
%U 302,306,312,326,356,366,380,382,400,416,422,426,452,460,486,490,496,500
%N Numbers n such that denominator of Sum_{k=0 to 2n+1} 1/k! is (2n+1)!/2.
%C The denominator of Sum_{k=0 to m} 1/k! is m!/d, where d = A093101(m). If m > 1 is odd, say m = 2n+1, then d is even. n is a member when d = 2. If m > 3 and m = 3 (mod 4), so that n > 1 is odd, then d is divisible by 4. So except for 1 the members are even.
%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="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.
%H <a href="/index/Fa#factorial">Index entries for sequences related to factorial numbers</a>
%F a(n) = 2*A102582(n-1) for n > 1.
%e 1/0! + 1/1! + 1/2! + 1/3! = 8/3 and 3 = (2*1+1)!/2, so 1 is a member.
%t fQ[n_] := (Denominator[ Sum[1/k!, {k, 0, 2n + 1}]] == (2n + 1)!/2); Select[ Range[ 500], fQ[ # ] &] (* _Robert G. Wilson v_, Jan 24 2005 *)
%Y n is a member <=> A093101(2n+1) = 2 <=> A061355(2n+1) = (2n+1)!/2 <=> n = 1 or n/2 is a member of A102582.
%K nonn
%O 1,2
%A _Jonathan Sondow_, Jan 21 2005
%E More terms from _Robert G. Wilson v_, Jan 24 2005
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