OFFSET
0,3
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
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.
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
EXAMPLE
a(4) = 2 because 2! <= 4 < 3!.
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 *)
PROG
(Haskell)
a084558 n = a090529 (n + 1) - 1 -- Reinhard Zumkeller, Jan 05 2014
(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)))) # Nathan L. Skirrow, May 28 2023
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Antti Karttunen, Jun 23 2003
EXTENSIONS
Name clarified by Antti Karttunen, Apr 27 2015
STATUS
approved