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

Offset: 1

Author

Clark Kimberling, Jul 18 2012

Keywords

• nonn
• easy

Comments

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

Examples

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

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.

Crossrefs

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

Programs

Mathematica

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