|
%I #37 Apr 13 2018 13:48:21
%S 1,2,5,8,65,163,1957,685,109601,98641,9864101,13563139,260412269,
%T 8463398743,47395032961,888656868019,56874039553217,7437374403113,
%U 17403456103284421,82666416490601,6613313319248080001,69439789852104840011
%N Numerator of Sum_{k=0..n} 1/k!.
%C p divides a(p-1) for prime p = {2, 5, 13, 37, 463, ...} which apparently coincides with A064384(n) = {2, 5, 13, 37, 463, ...} Primes p such that p divides 0!-1!+2!-3!+...+(-1)^{p-1}(p-1)!. - _Alexander Adamchuk_, Jun 14 2007
%C GCD(a(n), a(n+2)) = A124779(n) is either 1 or a prime 2, 5, 13, 37, 463, ... = A064384. - _Jonathan Sondow_, Jun 12 2007
%C For proofs of Adamchuk's and my Comments, see the link "The Taylor series for e ...". - _Jonathan Sondow_, Jun 18 2007
%H Harry J. Smith, <a href="/A061354/b061354.txt">Table of n, a(n) for n = 0..200</a>
%H J. Sondow, <a href="http://arxiv.org/abs/0704.1282">A geometric proof that e is irrational and a new measure of its irrationality</a>, Amer. Math. Monthly 113 (2006), 637-641.
%H J. Sondow, <a href="http://home.earthlink.net/~jsondow/PrimesAndE.pdf">The Taylor series for e and the primes 2, 5, 13, 37, 463, ...: a surprising connection</a>
%H 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>, 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 <a href="/index/Fa#factorial">Index entries for sequences related to factorial numbers</a>
%F a(n) = A000522(n)/A093101(n).
%F Numerators of floor(n!*exp(1))/n!, n>=1. Numerators of coefficients in expansion of exp(x)/(1-x). - _Vladeta Jovovic_, Aug 11 2002
%F a(n) = (1+n+n(n-1)+...+n!)/GCD(n!,1+n+n(n-1)+...+n!). - _Jonathan Sondow_, Aug 18 2006
%e 1, 2, 5/2, 8/3, 65/24, 163/60, 1957/720, 685/252, ...
%t exp[n_]:=Apply[Plus,1/Range[0,n]!];Numerator[Table[exp[n],{n,0,21}]] (* _Geoffrey Critzer_, May 05 2013 *)
%t A061354[n_] := Numerator[Sum[1/k!, {k, 0, n}]]; Array[A061354, 22, 0] (* _JungHwan Min_, Nov 08 2016 *)
%t Accumulate[1/Range[0,30]!]//Numerator (* _Harvey P. Dale_, Apr 13 2018 *)
%o (PARI) { default(realprecision, 500); e=exp(1); for (n=0, 200, a=numerator(floor(n!*e)/n!); if (n==0, a=1); write("b061354.txt", n, " ", a) ) } \\ _Harry J. Smith_, Jul 21 2009
%Y Cf. A061355 (denominators), A093101, A064384, A064384, A124779, A129924.
%K nonn,frac
%O 0,2
%A _Amarnath Murthy_, Apr 28 2001
|
