A003070 -id:A003070 - cp's OEIS frontend

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

Clark Kimberling, Jul 18 2012

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

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

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.

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

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

A269225 Smallest k such that k! > 2^n.

Original entry on oeis.org

2, 3, 3, 4, 4, 5, 5, 6, 6, 6, 7, 7, 7, 8, 8, 8, 9, 9, 9, 10, 10, 10, 11, 11, 11, 11, 12, 12, 12, 13, 13, 13, 13, 14, 14, 14, 14, 15, 15, 15, 15, 16, 16, 16, 16, 17, 17, 17, 17, 18, 18, 18, 18, 19, 19, 19, 19, 20, 20, 20, 20, 20, 21, 21, 21, 21, 22, 22, 22, 22, 23, 23, 23, 23, 23, 24, 24, 24, 24, 24, 25, 25, 25, 25, 26, 26, 26, 26, 26, 27, 27, 27, 27

Offset: 0

Christian Perfect, Jul 11 2016

			a(7) = 6 because 6! = 720 > 2^7 = 128, but 5! = 120 < 128.

Charles R Greathouse IV, Table of n, a(n) for n = 0..10000

Cf. A003070, A067850, A258782.

a[n_] := Block[{v=2^n, k=1}, While[++k! <= v]; k]; Array[a, 93, 0] (* Giovanni Resta, Jul 11 2016 *)
Module[{nn=30,f},f=Table[{k,k!},{k,nn}];Table[SelectFirst[f,#[[2]]>2^n&],{n,0,100}]][[;;,1]] (* Harvey P. Dale, Feb 19 2024 *)

a(n)=localprec(19); my(t=log(2)*n, x=ceil(solve(k=1, n/2+5, lngamma(k+1)-t))); while(x!<=2^n, x++); x \\ Charles R Greathouse IV, Jul 12 2016

def a269225():
   k = 1
   f = 1
   p = 1
   n = 0
   while True:
      while f<=p:
         k += 1
         f *= k
      yield k
      p *= 2
      n += 1

cp's OEIS Frontend

A214048 Least m>0 such that n! <= r^m, where r = (1+sqrt(5))/2, the golden ratio.

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