A090529 a(n) is the smallest positive m such that n <= m!.

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

Offset: 0

Author

Amarnath Murthy, Dec 07 2003

Keywords

• nonn

Comments

Define f(n,k) = floor[n/k]. Let f(n,2)= n_2, f(n_2,3) = n_3, ..., f(n_r, r+1) = n_(r+1). a(n) = least value of r such that n_r = 0. E.g., a(10) = 4, 10 -> 10/1 -> 10 -> 10/2 -> 5 -> 5/3 -> 1 -> 1/4 -> 0 in four steps.

Examples

			a(4)=3 because 2! < 4 <= 3!;
a(24)=4 because 3! < 24 <= 4!.
		

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.
• Index entries for sequences related to factorial numbers

Crossrefs

Cf. A084558, A000142.

Programs

Haskell

a090529 n = a090529_list !! n
a090529_list = f 1 1 0 where
  f w v u = if u <= w then v : f w v (u+1) else v' : f (w*v') v' (u+1)
            where v' = v + 1
-- Reinhard Zumkeller, Jan 05 2014

Maple

A090529 := proc(n)
    local a;
    for a from 1 do
        if a! >= n then
            return a;
        end if;
    end do:
end proc:
seq(A090529(n),n=0..100) ; # R. J. Mathar, Apr 06 2022

Mathematica

Array[Block[{m = 1}, While[# > m!, m++]; m] &, 105, 0] (* Michael De Vlieger, Nov 18 2017 *)
With[{f=Table[{m,m!},{m,10}]},Table[Select[f,#[[2]]>=n&][[1]],{n,0,110}]][[All,1]] (* Harvey P. Dale, Jan 01 2020 *)

PARI

a(n)=if(n<0,0,p=1;while(p!
				

PARI

A090529(n)=for(m=1,n,m!M. F. Hasler, Jan 16 2013

Formula

a(n+1) = A084558(n) + 1. - Reinhard Zumkeller, Jan 05 2014

Extensions

Better description and more terms from Zhang Wenpeng (wpzhang(AT)nwu.edu.cn), Mar 29 2004