A084558 a(0) = 0; for n >= 1: a(n) = largest m such that n >= m!.

0, 1, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5

Offset: 0

Author

Antti Karttunen, Jun 23 2003

Keywords

• nonn
• easy
• changed

Comments

For n >= 1, a(n) = the number of significant digits in n's factorial base representation (A007623).

After zero, which occurs once, each n occurs A001563(n) times.

Number of iterations (...(f_4(f_3(f_2(n))))...) such that the result is < 1, where f_j(x):=x/j. - Hieronymus Fischer, Apr 30 2012

For n > 0: a(n) = length of row n in table A108731. - Reinhard Zumkeller, Jan 05 2014

Examples

			a(4) = 2 because 2! <= 4 < 3!.
		

References

• F. Smarandache, "f-Inferior and f-Superior Functions - Generalization of Floor Functions", Arizona State University, Special Collections.

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
• Yi Yuan and Zhang Wenpeng, On the Mean Value of the Analogue of Smarandache Function, Smarandache Notions J., Vol. 15.

Crossrefs

A dual to A090529.

Cf. A084555, A084556, A084557.

Cf. A001069, A010096.

Cf. A000142, A001563, A055089, A060130, A111095, A211664, A211670, A108731, A212598, A220656, A220657, A220658, A220659, A231716, A235224, A257510.

Programs

Haskell

a084558 n = a090529 (n + 1) - 1  -- Reinhard Zumkeller, Jan 05 2014

Maple

0, seq(m$(m*m!),m=1..5); # Robert Israel, Apr 27 2015

Mathematica

Table[m = 1; While[m! <= n, m++]; m - 1, {n, 0, 104}] (* Jayanta Basu, May 24 2013 *)
Table[Floor[Last[Reduce[x! == n && x > 0, x]]], {n, 120}] (* Eric W. Weisstein, Sep 13 2024 *)

PARI

a(n)={my(m=0);while(n\=m++,);m-1} \\ R. J. Cano, Apr 09 2018

Python

def A084558(n):
  i=1
  while n: i+=1; n//=i
  return(i-1)
print(list(map(A084558,range(101)))) # Natalia L. Skirrow, May 28 2023

Formula

From Hieronymus Fischer, Apr 30 2012: (Start)

a(n!) = a((n-1)!)+1, for n>1.

G.f.: 1/(1-x)*Sum_{k>=1} x^(k!).

The explicit first terms of the g.f. are: (x+x^2+x^6+x^24+x^120+x^720...)/(1-x).

(End)

Other identities:

For all n >= 0, a(n) = A090529(n+1) - 1. - Reinhard Zumkeller, Jan 05 2014

For all n >= 1, a(n) = A060130(n) + A257510(n). - Antti Karttunen, Apr 27 2015

a(n) ~ log(n^2/(2*Pi)) / (2*LambertW(log(n^2/(2*Pi))/(2*exp(1)))) - 1/2. - Vaclav Kotesovec, Aug 22 2025

Extensions

Name clarified by Antti Karttunen, Apr 27 2015