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

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, 58, 61, 65, 68, 71, 75, 78, 82, 85, 89, 92, 96, 99, 103, 107, 110, 114, 118, 122, 125, 129, 133, 137, 141, 145, 148, 152, 156, 160, 164, 168, 172, 176, 180

Offset

0,4

Comments

a(n) is also the nearest integer to log(n!). - Eric M. Schmidt, Jun 19 2015

Log(n!) is asymptotic to A275341. - Mats Granvik, Aug 02 2016

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

References

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979, Section 22.1.

Links

Eric M. Schmidt, Table of n, a(n) for n = 0..10000 (corrected by Sean A. Irvine, Jan 18 2019)

Formula

a(n) = n*log(n) - n + O(log(n)). - Arkadiusz Wesolowski, Oct 18 2013

a(n) = round(LogGamma(n + 1)). - Mats Granvik, Roger L. Bagula, Aug 06 2016

a(n) = round(log(Product_{k=1..n} A139547(n,k))). - Mats Granvik, Aug 07 2016

Mathematica

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 *)

Prog

(Magma) [Round(Log(Factorial(n))): n in [2..100]]; // Vincenzo Librandi, Jun 19 2015
(PARI) A046654(n)=round(lngamma(n+1)) \\ M. F. Hasler, Dec 03 2018

Crossrefs

Cf. A025201.

Sequence in context: A095737 A054022 A185603 * A280724 A023543 A129895

Adjacent sequences: A046651 A046652 A046653 * A046655 A046656 A046657

Keyword

nonn

Author

N. J. A. Sloane, Dec 27 1999

Extensions

Name edited and a(0) = 0 prepended by M. F. Hasler, Dec 03 2018

Status

approved