

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
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



FORMULA

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



CROSSREFS



KEYWORD

nonn


AUTHOR



EXTENSIONS

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


STATUS

approved



