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

0,3

Comments

An inverse of A002034: A002034(a(n)) = n for n > 0. But not the least inverse: a(n) > A046021(n) for n > 3. - Jonathan Sondow, Jan 09 2005

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.

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)-1)). - Gerry Martens, May 31 2018

Example

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

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

Mathematica

A061355[n_] := Denominator[Sum[1/k!, {k, 0, n}]]; Array[A061355, 23, 0] (* JungHwan Min, Nov 08 2016 *)

Prog

(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
(GAP) List(List([0..25], n->Sum([0..n], k->1/Factorial(k))), DenominatorRat); # Muniru A Asiru, Jun 01 2018

Crossrefs

Cf. A061354 (numerators), A093101.

a(n) = n!/A093101(n) for n > 0. See also A002034, A046021.

Sequence in context: A061778 A160667 A118204 * A160618 A279335 A343566

Adjacent sequences: A061352 A061353 A061354 * A061356 A061357 A061358

Keyword

nonn,frac

Author

Amarnath Murthy, Apr 28 2001

Status

approved