A347009 a(n) is the largest integer m such that e^-m > e - Sum_{k=0..n} 1/k!.

-1, 0, 1, 2, 4, 6, 8, 10, 12, 15, 17, 19, 22, 25, 27, 30, 33, 36, 39, 42, 45, 48, 51, 54, 57, 61, 64, 67, 71, 74, 78, 81, 85, 88, 92, 95, 99, 102, 106, 110, 114, 117, 121, 125, 129, 132, 136, 140, 144, 148, 152, 156, 160, 164, 168, 172, 176, 180, 184, 188, 192

Offset

0,4

Comments

From Jon E. Schoenfield, Aug 11 2021: (Start)

It follows from the definition that a(n) = floor(-log(e - Sum_{k=0..n} 1/k!)) = floor(-log(Sum_{k>=n+1} 1/k!)) = floor(f(n)) where, for large n, f(n) = (n + 3/2)*log(n) - n - zeta'(0) + (1/12)/n + 1/n^2 - (361/360)/n^3 - (3/2)/n^4 + (10081/1260)/n^5 - (61/6)/n^6 - 40/n^7 - ...

Conjecture: a(n) = floor((n + 3/2)*log(n) - n - zeta'(0) + (1/12)/n + 1/n^2 for n >= 4.

(End)

Links

Table of n, a(n) for n=0..60.

Mathematica

Floor[Log[N[(1/(E - Sum[1/n!, {n, 0, #}] & /@ Range[50])), 2]]]

Crossrefs

Cf. A001113.

Sequence in context: A113903 A248563 A278453 * A130261 A186384 A011860

Adjacent sequences: A347006 A347007 A347008 * A347010 A347011 A347012

Keyword

sign

Author

Fred Patrick Doty, Aug 10 2021

Extensions

More terms from Jon E. Schoenfield, Aug 11 2021

Status

approved