A060151 Number of base n digits required to write n!.

1, 2, 2, 3, 3, 4, 5, 6, 6, 7, 8, 9, 9, 10, 11, 12, 12, 13, 14, 15, 15, 16, 17, 18, 19, 19, 20, 21, 22, 22, 23, 24, 25, 26, 26, 27, 28, 29, 30, 30, 31, 32, 33, 34, 34, 35, 36, 37, 38, 38, 39, 40, 41, 42, 43, 43, 44, 45, 46, 47, 47, 48, 49, 50, 51, 51, 52, 53, 54, 55, 56, 56, 57

Offset: 1

Author

Henry Bottomley, Mar 08 2001

Keywords

• base
• nonn
• easy

Examples

			a(6)=4 since 6!=720, which in base 6 is 3200.
		

Links

• Harry J. Smith and Danny Rorabaugh, Table of n, a(n) for n = 1..10000 (first 1000 terms from Harry J. Smith)

Crossrefs

Cf. A011776 for number of final zeros of n! written in base n.

Cf. A039960, A074181-A074184.

Programs

Magma

[1] cat [1 + Floor(Log(Factorial(n))/Log(n)): n in [2..80]]; // Vincenzo Librandi, Apr 15 2015

Mathematica

Join[{1},Table[IntegerLength[n!,n],{n,2,80}]] (* Harvey P. Dale, May 30 2014 *)

PARI

a(n)=if(n>1, logint(n!,n), 1) \\ Charles R Greathouse IV, Oct 29 2016

PARI

a(n)=if(n>1, lngamma(n+1)\log(n))+1 \\ Charles R Greathouse IV, Oct 29 2016

Sage

[1] + [1 + floor(log(factorial(n))/log(n)) for n in range(2,74)] # Danny Rorabaugh, Apr 14 2015

Formula

a(n) = 1 + floor(log(n!)/log(n)) = 1 + A039960(n) for n>1.

From Danny Rorabaugh, Apr 14 2015: (Start)

a(n) = 1 + log_n(A074182(n)) for n>1.

a(n) = A074184(n) = log_n(A074181(n)) for n>2.

(End)

a(n) = n - n/log n + O(1). - Charles R Greathouse IV, Oct 29 2016