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A046654 Nearest integer to Sum_{k=1..n} log(k) = log(n!). 9
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 (list; graph; refs; listen; history; text; internal format)
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
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

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Last modified June 20 18:51 EDT 2024. Contains 373532 sequences. (Running on oeis4.)