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A061354
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Numerator of Sum_{k=0..n} 1/k!.
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34
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1, 2, 5, 8, 65, 163, 1957, 685, 109601, 98641, 9864101, 13563139, 260412269, 8463398743, 47395032961, 888656868019, 56874039553217, 7437374403113, 17403456103284421, 82666416490601, 6613313319248080001, 69439789852104840011
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OFFSET
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0,2
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COMMENTS
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p divides a(p-1) for prime p = {2, 5, 13, 37, 463, ...} which apparently coincides with A064384(n) = {2, 5, 13, 37, 463, ...} Primes p such that p divides 0!-1!+2!-3!+...+(-1)^{p-1}(p-1)!. - Alexander Adamchuk, Jun 14 2007
For proofs of Adamchuk's and my Comments, see the link "The Taylor series for e ...". - Jonathan Sondow, Jun 18 2007
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LINKS
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J. Sondow and K. Schalm, 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, Gems in Experimental Mathematics (T. Amdeberhan, L. A. Medina, and V. H. Moll, eds.), Contemporary Mathematics, vol. 517, Amer. Math. Soc., Providence, RI, 2010.
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FORMULA
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Numerators of floor(n!*exp(1))/n!, n>=1. Numerators of coefficients in expansion of exp(x)/(1-x). - Vladeta Jovovic, Aug 11 2002
a(n) = (1+n+n(n-1)+...+n!)/GCD(n!,1+n+n(n-1)+...+n!). - Jonathan Sondow, Aug 18 2006
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EXAMPLE
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1, 2, 5/2, 8/3, 65/24, 163/60, 1957/720, 685/252, ...
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MATHEMATICA
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exp[n_]:=Apply[Plus, 1/Range[0, n]!]; Numerator[Table[exp[n], {n, 0, 21}]] (* Geoffrey Critzer, May 05 2013 *)
Accumulate[1/Range[0, 30]!]//Numerator (* Harvey P. Dale, Apr 13 2018 *)
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PROG
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(PARI) { default(realprecision, 500); e=exp(1); for (n=0, 200, a=numerator(floor(n!*e)/n!); if (n==0, a=1); write("b061354.txt", n, " ", a) ) } \\ Harry J. Smith, Jul 21 2009
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CROSSREFS
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KEYWORD
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nonn,frac
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AUTHOR
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STATUS
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approved
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