A025201 a(n) = floor(log(n!)).

0, 0, 1, 3, 4, 6, 8, 10, 12, 15, 17, 19, 22, 25, 27, 30, 33, 36, 39, 42, 45, 48, 51, 54, 58, 61, 64, 67, 71, 74, 78, 81, 85, 88, 92, 95, 99, 102, 106, 110, 114, 117, 121, 125, 129, 132, 136, 140, 144, 148, 152, 156, 160, 164, 168, 172, 176, 180, 184, 188, 192, 196, 201, 205, 209, 213, 217, 221, 226

Offset

1,4

Comments

Previous name was: a(n) = floor(sum_{k=1..n} log(k)). - Jon E. Schoenfield, Dec 29 2014

Logarithm of gamma function of n gives essentially the same sequence, only the index is shifted by 1. - Jason Earls, Jun 30 2001

a(n) is the largest integer m such that e^m <= n!. - Stanislav Sykora, May 29 2015

a(n) is within 1 of n*(log(n) - 1) + log(sqrt(2*Pi*n)). - Robert Israel, May 29 2015

Log(n!) ~ n log (n) - n + log(n(1+4n(1+2n)))/6 + log(Pi)/2 [given by Srinivasa Ramanujan, see Ramanujan's Lost Notebook, 1987]. - Robert G. Wilson v, Jun 11 2015

Robert Israel's formula above is Stirling's formula for log(n!). It gives the correct integer part for all n > 1 at least up to 10^6. It is known that the error is between 1/(12n+1) and 1/12n, therefore positive and decreasing to zero. However, log(n!) is never an integer and its fractional part appears to get arbitrarily close to any value in [0,1]. If ever the fractional part of log(n!) would be less than 1/(12n+1), then this approximation would yield a(n)-1 instead of a(n). - M. F. Hasler, Dec 03 2018

References

S. Ramanujan, The Lost Notebook and other Unpublished Papers. S. Raghavan and S. S. Rangachari, editors. Narosa, New Delhi, 1987, page 339.

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Wikipedia, Stirling's approximation: Speed of convergence and error estimates.

Formula

a(n) = floor(n*log(n) - n + log(2*Pi*n)/2) (Stirling's approximation) holds for all 1 < n < 10^6, but a counterexample might exist for some larger n. - M. F. Hasler, Dec 03 2018

Maple

seq(floor(lnGAMMA(n+1)), n=1..100); # Robert Israel, May 29 2015

Mathematica

Floor[ LogGamma[ Range[ 69] + 1]] (* Harvey P. Dale, Aug 23 2014 and slightly modified by Robert G. Wilson v, Jun 21 2015 to correct index *)
f[n_] := Floor[Log[n!]]; Array[f, 69] (* Robert G. Wilson v, Jun 11 2015 *)

Prog

(PARI) vector(100, n, floor(lngamma(n+1)) )  \\ Joerg Arndt, Dec 30 2014
(PARI) A025201(n)=floor(lngamma(n+1)) \\ floor(n*(log(n)-(n>1))+log(2*Pi*n)/2) is 2-3 times faster but could yield a(n)-1 instead of a(n), if frac(log(n!)) < 1/(12n+1). This doesn't happen up to n = 10^6, though. - M. F. Hasler, Dec 03 2018
(Magma) [Floor(Log(Factorial(n))): n in [1..100]]; // Vincenzo Librandi, May 30 2015

Crossrefs

Cf. A001113, A046654.

Sequence in context: A341340 A242932 A351863 * A071259 A231405 A092137

Adjacent sequences: A025198 A025199 A025200 * A025202 A025203 A025204

Keyword

nonn

Author

Clark Kimberling

Extensions

Simpler name (originally submitted in Formula section) from Mohammed Bouayoun (bouyao(AT)wanadoo.fr), Mar 20 2004 - Jon E. Schoenfield, Dec 29 2014

Status

approved