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

Author

N. J. A. Sloane, Dec 27 1999

Keywords

• nonn

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)

Crossrefs

Cf. A025201.

Programs

Magma

[Round(Log(Factorial(n))): n in [2..100]]; // Vincenzo Librandi, Jun 19 2015

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 *)

PARI

A046654(n)=round(lngamma(n+1)) \\ M. F. Hasler, Dec 03 2018

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

Extensions

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