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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!. 4
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 (list; graph; refs; listen; history; text; internal format)
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 A025261 A111572

Adjacent sequences:  A124779 A124780 A124781 * A124783 A124784 A124785

KEYWORD

nonn

AUTHOR

Jonathan Sondow, Nov 07 2006

STATUS

approved

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Last modified April 18 06:44 EDT 2019. Contains 322209 sequences. (Running on oeis4.)