A124781 a(n) = gcd(A093101(n), A093101(n+2)) where A093101(n) = gcd(n!, A(n)) and A(n) = A000522(n) = Sum_{k=0..n} n!/k!.

1, 1, 1, 2, 1, 2, 1, 10, 1, 2, 1, 2, 5, 2, 1, 2, 1, 10, 1, 2, 1, 2, 5, 26, 1, 2, 1, 10, 1, 2, 1, 2, 5, 2, 1, 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

0,4

Comments

a(n) divides n+3 because 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, 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

Formula

a(n) = gcd(A093101(n), A093101(n+2)) = (n+3)/A123901(n).

a(n) = gcd(A(n), A(n+2), n!) where A(n)=1+n+n(n-1)+...+n!. - Jonathan Sondow, Nov 13 2006

Example

a(3) = gcd(d(3),d(5)) = gcd(gcd(3!,16), gcd(5!,326)) = gcd(2,2) = 2.

Mathematica

(A[n_] := Sum[n!/k!, {k, 0, n}]; d[n_] := GCD[n!, A[n]]; Table[GCD[d[n], d[n+2]], {n, 0, 100}])
(* Second program, faster: *)
Table[GCD @@ Map[GCD[#!, Floor[E*#!] - Boole[# == 0]] &, n + {0, 2}], {n, 0, 96}] (* Michael De Vlieger, Jul 12 2017 *)

Prog

(PARI)
A000522(n) = sum(k=0, n, binomial(n, k)*k!); \\ This function from Joerg Arndt, Dec 14 2014
A093101(n) = gcd(n!, A000522(n));
m1=m2=1; for(n=0, 4096, m=m1; m1=m2; m2 = A093101(n+2); m124781 = gcd(m, m2); write("b093101.txt", n, " ", m); write("b124781.txt", n, " ", m124781); write("b123901.txt", n, " ", (n+3)/m124781)); \\ Antti Karttunen, Jul 12 2017

Crossrefs

Cf. A000522, A093101, A123899, A123900, A123901, A124779, A124780, A124782.

Sequence in context: A066772 A104060 A062347 * A124151 A110179 A071559

Adjacent sequences: A124778 A124779 A124780 * A124782 A124783 A124784

Keyword

nonn

Author

Jonathan Sondow, Nov 07 2006

Extensions

Replaced d(n) in the name with A093101(n). - Antti Karttunen, Jul 12 2017

Status

approved