A061355 Denominator of Sum_{k=0..n} 1/k!.
1, 1, 2, 3, 24, 60, 720, 252, 40320, 36288, 3628800, 4989600, 95800320, 3113510400, 17435658240, 326918592000, 20922789888000, 2736057139200, 6402373705728000, 30411275102208, 2432902008176640000, 25545471085854720000, 224800145555521536000
Offset: 0
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, arXiv:0704.1282 [math.HO], 2007-2010.
- 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 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, arXiv:0709.0671 [math.NT], 2007-2009; 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
Programs
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GAP
List(List([0..25],n->Sum([0..n],k->1/Factorial(k))),DenominatorRat); # Muniru A Asiru, Jun 01 2018
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Maple
BB:=n->sum(1/i!, i=1..n): a:=n->floor(denom(BB(n))): seq(a(n), n=0..22); # Zerinvary Lajos, Mar 28 2007
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Mathematica
A061355[n_] := Denominator[Sum[1/k!, {k, 0, n}]]; Array[A061355, 23, 0] (* JungHwan Min, Nov 08 2016 *) Accumulate[1/Range[0,30]!]//Denominator (* Harvey P. Dale, Mar 24 2025 *)
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PARI
{ default(realprecision, 500); e=exp(1); for (n=0, 200, a=denominator(floor(n!*e)/n!); write("b061355.txt", n, " ", a) ) } \\ Harry J. Smith, Jul 21 2009 -
PARI
first(n) = my(res = vector(n), s = 0, f = 1); for(i = 1, n, f *= i; s += 1/f; res[i] = denominator(s)); res \\ David A. Corneth, May 31 2018
Formula
Denominators of floor(n!*exp(1))/n!. Denominators of coefficients in expansion of exp(x)/(1-x). - Vladeta Jovovic, Aug 11 2002
a(n) = n!/gcd(n!, 1 + n + n(n-1) + n(n-1)(n-2) + ... + n!). - Jonathan Sondow, Jan 09 2005
a(n) = denominator(exp(1)*gamma(n + 1,1)/gamma(n + 1)). - Gerry Martens, May 31 2018
Comments