A214048 - OEIS

A214048

Least m>0 such that n! <= r^m, where r = (1+sqrt(5))/2, the golden ratio.

1, 2, 4, 7, 10, 14, 18, 23, 27, 32, 37, 42, 47, 53, 58, 64, 70, 76, 82, 88, 95, 101, 108, 114, 121, 128, 135, 142, 149, 156, 163, 170, 177, 185, 192, 199, 207, 214, 222, 230, 237, 245, 253, 261, 269, 277, 285, 293, 301, 309

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OFFSET

1,2

COMMENTS

Also, the least m>0 such that n! < L(m), where L = A000032, the Lucas numbers.

LINKS

Clark Kimberling, Table of n, a(n) for n = 1..1000

Aleksandar Petojević, Lambert's W function and Kurepa's left factorial, Project: Kurepa's hypothesis for left factorial, ResearchGate (2023).

Aleksandar Petojević, Marjana Gorjanac Ranitović, Dragan Rastovac, and Milinko Mandić, The Golden Ratio, Factorials, and the Lambert W Function, Journal of Integer Sequences, Vol. 27 (2024), Article 24.5.7.

EXAMPLE

a(4) = 7 because r^6 < 4! <= 4^7.

MATHEMATICA

Table[m=1; While[n!>GoldenRatio^m, m++]; m, {n, 1, 100}]

CROSSREFS

Cf. A000032, A001622, A003070, A213857, A214047.

Sequence in context: A225635 A256967 A225249 * A328659 A088236 A194244

Adjacent sequences: A214045 A214046 A214047 * A214049 A214050 A214051

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling, Jul 18 2012

STATUS

approved