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Denominators of certain upper bounds for Euler's number e.
1

%I #12 Apr 29 2023 23:01:41

%S 1,4,18,32,600,4320,11760,322560,3265920,1728000,439084800,821145600,

%T 817689600,1220496076800,19615115520000,111588212736000,

%U 863812325376000,115242726703104000,15722836107264000,3742926166425600000

%N Denominators of certain upper bounds for Euler's number e.

%C For the numerators see A095822.

%C e := Sum_{k>=0} 1/k! has (trivial) upper bound r(n) := A095822(n)/a(n), for every n >= 1. See the W. Lang link.

%D M. Barner and F. Flohr, Analysis I, de Gruyter, 5te Auflage, 2000; pp. 117/8.

%D E. Kuz'min and A. I. Shirshov: On the number e, pp. 111-119, eq.(6), in: Kvant Selecta: Algebra and Analysis, I, ed. S. Tabachnikov, Am.Math.Soc., 1999

%H W. Lang, <a href="/A095822/a095822.txt">r(n) numbers and comments</a>.

%F a(n) = denominator(r(n)), with rational r(n) := (Sum_{k=0..n} 1/k!) + 1/(n*n!), n >= 1, written in lowest terms. For n*n! see A001563(n).

%e The positive rationals r(n), n >= 1: 3/1, 11/4, 49/18, 87/32, 1631/600, 11743/4320, 31967/11760, ...

%K nonn,easy,frac

%O 1,2

%A _Wolfdieter Lang_, Jun 11 2004