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%I #61 Sep 08 2022 08:44:48
%S 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,
%T 61,64,67,71,74,78,81,85,88,92,95,99,102,106,110,114,117,121,125,129,
%U 132,136,140,144,148,152,156,160,164,168,172,176,180,184,188,192,196,201,205,209,213,217,221,226
%N a(n) = floor(log(n!)).
%C Previous name was: a(n) = floor(sum_{k=1..n} log(k)). - _Jon E. Schoenfield_, Dec 29 2014
%C Logarithm of gamma function of n gives essentially the same sequence, only the index is shifted by 1. - _Jason Earls_, Jun 30 2001
%C a(n) is the largest integer m such that e^m <= n!. - _Stanislav Sykora_, May 29 2015
%C a(n) is within 1 of n*(log(n) - 1) + log(sqrt(2*Pi*n)). - _Robert Israel_, May 29 2015
%C 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
%C 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
%D S. Ramanujan, The Lost Notebook and other Unpublished Papers. S. Raghavan and S. S. Rangachari, editors. Narosa, New Delhi, 1987, page 339.
%H Robert Israel, <a href="/A025201/b025201.txt">Table of n, a(n) for n = 1..10000</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Stirling%27s_approximation#Speed_of_convergence_and_error_estimates">Stirling's approximation: Speed of convergence and error estimates</a>.
%F 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
%p seq(floor(lnGAMMA(n+1)),n=1..100);# _Robert Israel_, May 29 2015
%t 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 *)
%t f[n_] := Floor[Log[n!]]; Array[f, 69] (* _Robert G. Wilson v_, Jun 11 2015 *)
%o (PARI) vector(100, n, floor(lngamma(n+1)) ) \\ _Joerg Arndt_, Dec 30 2014
%o (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
%o (Magma) [Floor(Log(Factorial(n))): n in [1..100]]; // _Vincenzo Librandi_, May 30 2015
%Y Cf. A001113, A046654.
%K nonn
%O 1,4
%A _Clark Kimberling_
%E Simpler name (originally submitted in Formula section) from Mohammed Bouayoun (bouyao(AT)wanadoo.fr), Mar 20 2004 - _Jon E. Schoenfield_, Dec 29 2014
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