A061354 Numerator of Sum_{k=0..n} 1/k!.

1, 2, 5, 8, 65, 163, 1957, 685, 109601, 98641, 9864101, 13563139, 260412269, 8463398743, 47395032961, 888656868019, 56874039553217, 7437374403113, 17403456103284421, 82666416490601, 6613313319248080001, 69439789852104840011

Offset: 0

Author

Amarnath Murthy, Apr 28 2001

Keywords

• nonn
• frac

Comments

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

GCD(a(n), a(n+2)) = A124779(n) is either 1 or a prime 2, 5, 13, 37, 463, ... = A064384. - Jonathan Sondow, Jun 12 2007

For proofs of Adamchuk's and my Comments, see the link "The Taylor series for e ...". - Jonathan Sondow, Jun 18 2007

Examples

			1, 2, 5/2, 8/3, 65/24, 163/60, 1957/720, 685/252, ...
		

Links

• Harry J. Smith, Table of n, a(n) for n = 0..200
• J. Sondow, A geometric proof that e is irrational and a new measure of its irrationality, Amer. Math. Monthly 113 (2006), 637-641.
• J. Sondow, The Taylor series for e and the primes 2, 5, 13, 37, 463, ...: a surprising connection
• 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.
• Index entries for sequences related to factorial numbers

Crossrefs

Cf. A061355 (denominators), A093101, A064384, A064384, A124779, A129924.

Programs

Mathematica

exp[n_]:=Apply[Plus,1/Range[0,n]!];Numerator[Table[exp[n],{n,0,21}]]  (* Geoffrey Critzer, May 05 2013 *)
A061354[n_] := Numerator[Sum[1/k!, {k, 0, n}]]; Array[A061354, 22, 0] (* JungHwan Min, Nov 08 2016 *)
Accumulate[1/Range[0,30]!]//Numerator (* Harvey P. Dale, Apr 13 2018 *)

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

Formula

a(n) = A000522(n)/A093101(n).

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