A124782 a(n) = (n+3)/gcd(A(n), A(n+2)) where A(n) = A000522(n) = Sum_{k=0..n} n!/k!.

3, 2, 1, 3, 7, 4, 9, 1, 11, 6, 1, 7, 3, 8, 17, 9, 19, 2, 21, 11, 23, 12, 5, 1, 27, 14, 29, 3, 31, 16, 33, 17, 7, 18, 1, 19, 3, 4, 41, 21, 43, 22, 9, 23, 47, 24, 49, 5, 51, 2, 53, 27, 11, 28, 57, 29, 59, 6, 61, 31, 63, 32, 1, 33, 67, 34, 69, 7, 71, 36, 73, 1, 15, 38, 77, 3, 79, 8, 81, 41

Offset

0,1

Comments

a(n) is an integer since A(n+2) = (n+2)(n+1)*A(n) + n+3.

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, [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

Formula

a(n) = (n+3)/A124780(n) = (n+3)/gcd(A000522(n), A000522(n+2)).

Example

a(3) = (3+3)/gcd(A(3), A(5)) = 6/gcd(16, 326) = 6/2 = 3.

Mathematica

(A[n_] := Sum[n!/k!, {k, 0, n}]; Table[(n+3)/GCD[A[n], A[n+2]], {n, 0, 80}])

Prog

(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));
A124782(n) = ((n+3)/A124780(n)); \\ Antti Karttunen, Jul 07 2017

Crossrefs

Cf. A000522, A093101, A123899, A123900, A123901, A124779, A124780, A124781.

Sequence in context: A097409 A257556 A078268 * A106611 A331523 A025261

Adjacent sequences: A124779 A124780 A124781 * A124783 A124784 A124785

Keyword

nonn

Author

Jonathan Sondow, Nov 07 2006

Status

approved