A138761 - OEIS

A138761

a(n) is the smallest member of A000522 divisible by 2^n, where A000522(m) = total number of arrangements of a set with m elements.

1, 2, 16, 16, 16, 330665665962404000, 4216377920843140187197325631474390438452208808916276571342090223552

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OFFSET

0,2

COMMENTS

a(n) < A000522(2^n) for n > 0; see Sondow and Schalm, Proposition A.13 part (ii).

REFERENCES

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), Gems in Experimental Mathematics (T. Amdeberhan, L. A. Medina, and V. H. Moll, eds.), Contemporary Mathematics, vol. 517, Amer. Math. Soc., Providence, RI, 2010.

LINKS

Jean-François Alcover, Table of n, a(n) for n = 0..9

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, arXiv:0709.0671 [math.NT], 2007-2009.

Index entries for sequences related to factorial numbers

FORMULA

a(n) = A000522(A127014(n)) = Sum_{k=0..A127014(n)} A127014(n)!/k! for n > 0.

EXAMPLE

a(5) = A000522(19) = 330665665962404000 because that is the smallest member of A000522 divisible by 2^5.

MATHEMATICA

a522[n_] := E Gamma[n + 1, 1];

(* b = A127014 *)

b[1] = 1; b[n_] := b[n] = For[k = b[n - 1], True, k++, If[Mod[a522[k], 2^n] == 0, Return[k]]];