A258782 Nearest integer to log_2(n!).
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, 84, 88, 93, 98, 103, 108, 113, 118, 123, 128, 133, 138, 143, 149, 154, 159, 165, 170, 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
Offset: 0
Examples
a(6) = round(log_2(6!)) = round(9.49...) = 9.
Programs
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MATLAB
for i = 1:20 { disp(round(log2(factorial(i)))) } end -
Magma
[Round(LogGamma(n+1)/Log(2)): n in [0..70]]; // Bruno Berselli, Jun 23 2015
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Maple
seq(round(lnGAMMA(n+1)/ln(2)),n=0..100); # Robert Israel, Jun 10 2015
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Mathematica
Round[Log[2, Range[0, 100]! ]] (* Giovanni Resta, Jun 10 2015 *)
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PARI
a(n) = round(log(n!)/log(2)); \\ Michel Marcus, Jun 10 2015
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PARI
a(n)=round(lngamma(n+1)/log(2)) \\ Charles R Greathouse IV, Jun 10 2015
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Sage
[round(log_gamma(n+1)/log2) for n in (0..70)] # Bruno Berselli, Jun 23 2015
Formula
a(n) = round(log_2(n!)).
a(n) = round(Sum_{k=1..n} log_2(k)). - Tom Edgar, Jun 10 2015
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