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

1,2

Links

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

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

Example

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

Mathematica

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

Prog

(PARI) default(realprecision, 10); for (n=1, 1000, if (n==1, a=1, a=1 + floor(log(n!)/log(n))); write("b060151.txt", n, " ", a)) \\ Harry J. Smith, Jul 02 2009
(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
(Magma) [1] cat [1 + Floor(Log(Factorial(n))/Log(n)): n in [2..80]]; // Vincenzo Librandi, Apr 15 2015

Crossrefs

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

Cf. A039960, A074181-A074184.

Sequence in context: A156253 A346693 A265436 * A285902 A097330 A086861

Adjacent sequences: A060148 A060149 A060150 * A060152 A060153 A060154

Keyword

base,nonn,easy

Author

Henry Bottomley, Mar 08 2001

Status

approved