A093101 Cancellation factor in reducing Sum_{k=0...n} 1/k! to lowest terms.

1, 1, 1, 2, 1, 2, 1, 20, 1, 10, 1, 8, 5, 2, 5, 4, 1, 130, 1, 4000, 1, 2, 5, 52, 5, 494, 1, 40, 1, 10, 13, 4, 25, 38, 5, 16, 13, 230, 13, 20, 1, 46, 5, 104, 475, 62, 1, 20, 1, 130, 31, 832, 2755, 74, 5, 4, 13, 50, 1, 40, 23, 2, 2795, 76, 34385, 2, 1, 80, 1, 650, 1, 2812, 5, 74, 5

Offset

0,4

Comments

Same as n!/A061355(n) and (1+n+n(n-1)+n(n-1)(n-2)+...+n!)/A061354(n).

a(n) is relatively prime to n.

gcd(a(n),a(n+1)) = 1.

Links

Antti Karttunen, Table of n, a(n) for n = 0..4096

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, A geometric proof that e is irrational and a new measure of its irrationality, arXiv:0704.1282 [math.HO], 2007-2010.

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.

Formula

a(n) = gcd(n!, 1+n+n(n-1)+n(n-1)(n-2)+...+n!).

a(n) = gcd(n!, A(n)) where A(0) = 1, A(n) = n*A(n-1)+1.

Example

E.g. 1/0!+1/1!+1/2!+1/3!=16/6=(2*8)/(2*3) so a(3)=2.

Mathematica

f[n_] := n! / Denominator[ Sum[1/k!, {k, 0, n}]]; Table[ f[n], {n, 0, 74}] (* Robert G. Wilson v *)
(* Second program: *)
A[n_] := If[n==0, 1, n*A[n-1]+1]; Table[GCD[A[n], n! ], {n, 0, 74}]

Prog

(PARI)
A000522(n) = sum(k=0, n, binomial(n, k)*k!); \\ This function from Joerg Arndt, Dec 14 2014
A093101(n) = gcd(n!, A000522(n)); \\ Antti Karttunen, Jul 12 2017

Crossrefs

Cf. A093647, A093651.

(n+1)!/(a(n)*a(n+1)) = A123899(n).

(n+3)!/(a(n)*a(n+1)*a(n+2)) = A123900(n).

(n+3)/GCD(a(n), a(n+2)) = A123901(n).

Cf. also A000522, A061354, A061355.

Sequence in context: A287541 A288196 A072883 * A082469 A206566 A376596

Adjacent sequences: A093098 A093099 A093100 * A093102 A093103 A093104

Keyword

nonn

Author

Jonathan Sondow, May 10 2004, Oct 18 2006

Extensions

More terms from Robert G. Wilson v, May 14 2004

Status

approved