A124780 a(n) = gcd(A(n), A(n+2)) where A(n) = A000522(n) = Sum_{k=0..n} n!/k!.

1, 2, 5, 2, 1, 2, 1, 10, 1, 2, 13, 2, 5, 2, 1, 2, 1, 10, 1, 2, 1, 2, 5, 26, 1, 2, 1, 10, 1, 2, 1, 2, 5, 2, 37, 2, 13, 10, 1, 2, 1, 2, 5, 2, 1, 2, 1, 10, 1, 26, 1, 2, 5, 2, 1, 2, 1, 10, 1, 2, 1, 2, 65, 2, 1, 2, 1, 10, 1, 2, 1, 74, 5, 2, 1, 26, 1, 10, 1, 2, 1, 2, 5, 2, 1, 2, 1, 10, 13, 2, 1, 2, 5, 2, 1, 2, 1

Offset: 0

Author

Jonathan Sondow, Nov 07 2006

Keywords

• nonn

Comments

a(n) divides n+3 because A(n+2) = (n+2)(n+1)*A(n) + n+3.

Examples

			a(2) = gcd(A(2), A(4)) = gcd(5, 65) = 5.
		

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.
• Index entries for sequences related to factorial numbers

Crossrefs

Cf. A000522, A093101, A123899, A123900, A123901, A124779, A124781, A124782.

Programs

Mathematica

(A[n_] := Sum[n!/k!, {k, 0, n}]; Table[GCD[A[n],A[n+2]], {n,0,100}])
GCD[#[[1]],#[[3]]]&/@Partition[Table[Sum[n!/k!,{k,0,n}],{n,0,100}],3,1] (* Harvey P. Dale, Jun 14 2022 *)

PARI

A000522(n) = sum(k=0, n, binomial(n, k)*k!); \\ This function from Joerg Arndt, Dec 14 2014
A124780(n) = gcd(A000522(n),A000522(n+2)); \\ Antti Karttunen, Jul 07 2017

Formula

a(n) = gcd(A000522(n), A000522(n+2)) = (n+3)/A124782(n)