%I #20 Jan 10 2024 16:20:58
%S 1,2,5,20,101,620,4420,35894,326946,3301574,36613057,442369756,
%T 5784470466,81391912093,1226260443926,19696254286261,335987466998509,
%U 6066332690596289,115577941857034741,2317310520602816401,48773396185794559169,1075223007090667361164
%N Ceiling(1/(e - 1/0! - 1/1! - 1/2! - ... - 1/n!)).
%C a(n) = least k such that 1/k > e - (n-th partial sum of the Maclaurin series for e). Let b(n) = a(n)/a(n+1). Conjectures: if n > 3, then n+1 < b(n) < n+2 and 0 < b(n+1)-b(n) < 1.
%H Clark Kimberling, <a href="/A323274/b323274.txt">Table of n, a(n) for n = 0..1000</a>
%e Approximates for the first 5 numbers e - (1/0!+1/1!+1/2!+...+1/n! are 1.71828, 0.718282, 0.218282, 0.0516152, 0.0099485, with approximate reciprocals 0.581977, 1.39221, 4.58123, 19.3742, 100.518.
%t s[n_] := E - Sum[1/k!, {k, 0, n}]
%t Table[Ceiling[1/s[n]], {n, 0, 30}]
%Y Cf. A000142, A001113, A061354.
%K nonn
%O 0,2
%A _Clark Kimberling_, Jan 11 2019
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