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A214048
Least m>0 such that n! <= r^m, where r = (1+sqrt(5))/2, the golden ratio.
1
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
1,2
COMMENTS
Also, the least m>0 such that n! < L(m), where L = A000032, the Lucas numbers.
LINKS
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
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Jul 18 2012
STATUS
approved