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A084558
a(0) = 0; for n >= 1: a(n) = largest m such that n >= m!.
60
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
0,3
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
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
For all n >= 1, a(n) = A060130(n) + A257510(n). - Antti Karttunen, Apr 27 2015
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
KEYWORD
nonn,easy
AUTHOR
Antti Karttunen, Jun 23 2003
EXTENSIONS
Name clarified by Antti Karttunen, Apr 27 2015
STATUS
approved