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

0,3

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.

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

Formula

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

Example

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

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 *)

Prog

(PARI) a(n)=if(n<0, 0, p=1; while(p!<n, p++); p)
(PARI) A090529(n)=for(m=1, n, m!<n || return(m))  \\ M. F. Hasler, Jan 16 2013
(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

Crossrefs

Cf. A084558, A000142.

Sequence in context: A087162 A288914 A046925 * A297212 A155934 A251719

Adjacent sequences: A090526 A090527 A090528 * A090530 A090531 A090532

Keyword

nonn

Author

Amarnath Murthy, Dec 07 2003

Extensions

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

Status

approved