A258782 - OEIS

%I #41 Sep 08 2022 08:46:12

%S 0,0,1,3,5,7,9,12,15,18,22,25,29,33,36,40,44,48,53,57,61,65,70,74,79,

%T 84,88,93,98,103,108,113,118,123,128,133,138,143,149,154,159,165,170,

%U 175,181,186,192,197,203,209,214,220,226,231,237,243,249,254,260,266,272,278,284,290,296,302,308,314

%N Nearest integer to log_2(n!).

%F a(n) = round(log_2(n!)).

%F a(n) = A004257(A000142(n)). - _Michel Marcus_, Jun 10 2015

%F a(n) = round(Sum_{k=1..n} log_2(k)). - _Tom Edgar_, Jun 10 2015

%F a(n) is within 1 of n*(log(n)-1)/log(2) + log(n)/(2*log(2)) + log(sqrt(2*Pi))/log(2) for n >= 1. - _Robert Israel_, Jun 10 2015

%e a(6) = round(log_2(6!)) = round(9.49...) = 9.

%p seq(round(lnGAMMA(n+1)/ln(2)),n=0..100); # _Robert Israel_, Jun 10 2015

%t Round[Log[2, Range[0, 100]! ]] (* _Giovanni Resta_, Jun 10 2015 *)

%o (MATLAB) for i = 1:20 { disp(round(log2(factorial(i)))) } end

%o (PARI) a(n) = round(log(n!)/log(2)); \\ _Michel Marcus_, Jun 10 2015

%o (PARI) a(n)=round(lngamma(n+1)/log(2)) \\ _Charles R Greathouse IV_, Jun 10 2015

%o (Magma) [Round(LogGamma(n+1)/Log(2)): n in [0..70]]; // _Bruno Berselli_, Jun 23 2015

%o (Sage) [round(log_gamma(n+1)/log2) for n in (0..70)] # _Bruno Berselli_, Jun 23 2015

%Y Cf. A025201, A067850.

%K nonn,easy

%O 0,4

%A _Eli Sadoff_, Jun 10 2015