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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. 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.)