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A046654 Nearest integer to Sum_{k=1..n} log(k) = log(n!).

%I A046654 #39 Sep 08 2022 08:44:56
%S A046654 0,0,1,2,3,5,7,9,11,13,15,18,20,23,25,28,31,34,36,39,42,45,48,52,55,
%T A046654 58,61,65,68,71,75,78,82,85,89,92,96,99,103,107,110,114,118,122,125,
%U A046654 129,133,137,141,145,148,152,156,160,164,168,172,176,180
%N A046654 Nearest integer to Sum_{k=1..n} log(k) = log(n!).
%C A046654 a(n) is also the nearest integer to log(n!). - _Eric M. Schmidt_, Jun 19 2015
%C A046654 Log(n!) is asymptotic to A275341. - _Mats Granvik_, Aug 02 2016
%C A046654 Stirling's approximation s(n) = n*log(n) - n + log(2*Pi*n)/2 is known to be equal to log(n!) up to an error between 1/(12n + 1) and 1/12n. For all 0 < n < 10^6 except for n = 11, round(s(n)) = a(n). What is the next such exceptional index n? - _M. F. Hasler_, Dec 03 2018
%D A046654 G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979, Section 22.1.
%H A046654 Eric M. Schmidt, <a href="/A046654/b046654.txt">Table of n, a(n) for n = 0..10000</a> (corrected by Sean A. Irvine, Jan 18 2019)
%F A046654 a(n) = n*log(n) - n + O(log(n)). - _Arkadiusz Wesolowski_, Oct 18 2013
%F A046654 a(n) = round(LogGamma(n + 1)). - _Mats Granvik_, Roger L. Bagula, Aug 06 2016
%F A046654 a(n) = round(log(Product_{k=1..n} A139547(n,k))). - _Mats Granvik_, Aug 07 2016
%t A046654 nn = 58; t = Accumulate[Log /@ Range[nn]]; Table[If[(y = Ceiling[x = t[[i]]]) - x <= x - (z = Floor[x]), a = y, a = z]; a, {i, nn}] (* _Jayanta Basu_, Jun 27 2013 *)
%o A046654 (Magma) [Round(Log(Factorial(n))): n in [2..100]]; // _Vincenzo Librandi_, Jun 19 2015
%o A046654 (PARI) A046654(n)=round(lngamma(n+1)) \\ _M. F. Hasler_, Dec 03 2018
%Y A046654 Cf. A025201.
%K A046654 nonn
%O A046654 0,4
%A A046654 _N. J. A. Sloane_, Dec 27 1999
%E A046654 Name edited and a(0) = 0 prepended by _M. F. Hasler_, Dec 03 2018