A323274 a(n) = ceiling(1/(e - 1/0! - 1/1! - 1/2! - ... - 1/n!)).

1, 2, 5, 20, 101, 620, 4420, 35894, 326946, 3301574, 36613057, 442369756, 5784470466, 81391912093, 1226260443926, 19696254286261, 335987466998509, 6066332690596289, 115577941857034741, 2317310520602816401, 48773396185794559169, 1075223007090667361164

Offset: 0

Author

Clark Kimberling, Jan 11 2019

Keywords

• nonn

Comments

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.

Examples

			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.
		

Links

• Clark Kimberling, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A000142, A001113, A061354.

Programs

Mathematica

s[n_] := E - Sum[1/k!, {k, 0, n}]
Table[Ceiling[1/s[n]], {n, 0, 30}]