

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
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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



