A124782 - OEIS

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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

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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