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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.

%I A138761 #22 Feb 21 2019 03:06:21
%S A138761 1,2,16,16,16,330665665962404000,
%T A138761 4216377920843140187197325631474390438452208808916276571342090223552
%N 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.
%C A138761 a(n) < A000522(2^n) for n > 0; see Sondow and Schalm, Proposition A.13 part (ii).
%D A138761 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.
%H A138761 Jean-François Alcover, <a href="/A138761/b138761.txt">Table of n, a(n) for n = 0..9</a>
%H A138761 J. Sondow and K. Schalm, <a href="http://arxiv.org/abs/0709.0671">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</a>, arXiv:0709.0671 [math.NT], 2007-2009.
%H A138761 <a href="/index/Fa#factorial">Index entries for sequences related to factorial numbers</a>
%F A138761 a(n) = A000522(A127014(n)) = Sum_{k=0..A127014(n)} A127014(n)!/k! for n > 0.
%e A138761 a(5) = A000522(19) = 330665665962404000 because that is the smallest member of A000522 divisible by 2^5.
%t A138761 a522[n_] := E Gamma[n + 1, 1];
%t A138761 (* b = A127014 *)
%t A138761 b[1] = 1; b[n_] := b[n] = For[k = b[n - 1], True, k++, If[Mod[a522[k], 2^n] == 0, Return[k]]];
%t A138761 a[0] = 1; a[n_] := a522[b[n]];
%t A138761 Table[a[n], {n, 0, 6}] (* _Jean-François Alcover_, Feb 20 2019 *)
%Y A138761 Cf. A000522, A127014, A127015.
%K A138761 nonn
%O A138761 0,2
%A A138761 _Jonathan Sondow_, Apr 01 2008