A067850 -id:A067850 - cp's OEIS frontend

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

Eli Sadoff, Jun 10 2015

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

Cf. A025201, A067850.

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

[Round(LogGamma(n+1)/Log(2)): n in [0..70]]; // Bruno Berselli, Jun 23 2015

seq(round(lnGAMMA(n+1)/ln(2)),n=0..100); # Robert Israel, Jun 10 2015

Round[Log[2, Range[0, 100]! ]] (* Giovanni Resta, Jun 10 2015 *)

a(n) = round(log(n!)/log(2)); \\ Michel Marcus, Jun 10 2015

a(n)=round(lngamma(n+1)/log(2)) \\ Charles R Greathouse IV, Jun 10 2015

[round(log_gamma(n+1)/log2) for n in (0..70)] # Bruno Berselli, Jun 23 2015

a(n) = round(log_2(n!)).
a(n) = A004257(A000142(n)). - Michel Marcus, Jun 10 2015
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

A269225 Smallest k such that k! > 2^n.

Original entry on oeis.org

2, 3, 3, 4, 4, 5, 5, 6, 6, 6, 7, 7, 7, 8, 8, 8, 9, 9, 9, 10, 10, 10, 11, 11, 11, 11, 12, 12, 12, 13, 13, 13, 13, 14, 14, 14, 14, 15, 15, 15, 15, 16, 16, 16, 16, 17, 17, 17, 17, 18, 18, 18, 18, 19, 19, 19, 19, 20, 20, 20, 20, 20, 21, 21, 21, 21, 22, 22, 22, 22, 23, 23, 23, 23, 23, 24, 24, 24, 24, 24, 25, 25, 25, 25, 26, 26, 26, 26, 26, 27, 27, 27, 27

Offset: 0

Christian Perfect, Jul 11 2016

			a(7) = 6 because 6! = 720 > 2^7 = 128, but 5! = 120 < 128.

Charles R Greathouse IV, Table of n, a(n) for n = 0..10000

Cf. A003070, A067850, A258782.

a[n_] := Block[{v=2^n, k=1}, While[++k! <= v]; k]; Array[a, 93, 0] (* Giovanni Resta, Jul 11 2016 *)
Module[{nn=30,f},f=Table[{k,k!},{k,nn}];Table[SelectFirst[f,#[[2]]>2^n&],{n,0,100}]][[;;,1]] (* Harvey P. Dale, Feb 19 2024 *)

a(n)=localprec(19); my(t=log(2)*n, x=ceil(solve(k=1, n/2+5, lngamma(k+1)-t))); while(x!<=2^n, x++); x \\ Charles R Greathouse IV, Jul 12 2016

def a269225():
   k = 1
   f = 1
   p = 1
   n = 0
   while True:
      while f<=p:
         k += 1
         f *= k
      yield k
      p *= 2
      n += 1

A127038 Maximal value of m such that 23^m <= n! : a(n) = floor( log(n!) / log(23) ).

Original entry on oeis.org

0, 0, 0, 1, 1, 2, 2, 3, 4, 4, 5, 6, 7, 8, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 26, 27, 28, 29, 30, 31, 32, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44, 46, 47, 48, 49, 51, 52, 53, 54, 56, 57, 58, 60

Offset: 1

Artur Jasinski, Jan 03 2007

nonn

Cf. A067850, A127031, A127032, A127033, A127034, A127035, A127036, A127037, A127039.

a(n)=lngamma(n+1)\log(23) \\ Charles R Greathouse IV, Sep 04 2015

a(n)=logint(n!,23) \\ Charles R Greathouse IV, Sep 04 2015

A127041 Maximal value of m such that 31^m <= n! : a(n) = floor( log(n!) / log(31) ).

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 2, 3, 3, 4, 5, 5, 6, 7, 8, 8, 9, 10, 11, 12, 13, 14, 15, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 54

Offset: 1

Artur Jasinski, Jan 03 2007

nonn

Cf. A067850, A127031, A127032, A127033, A127034, A127035, A127036, A127037, A127039.

a(n) =  floor( log(n!) / log(31) ); \\ Michel Marcus, Nov 02 2013

a(n)=lngamma(n+1)\log(31) \\ Charles R Greathouse IV, Sep 04 2015

a(n)=logint(n!,31) \\ Charles R Greathouse IV, Sep 04 2015

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A258782 Nearest integer to log_2(n!).

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A269225 Smallest k such that k! > 2^n.

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A127038 Maximal value of m such that 23^m <= n! : a(n) = floor( log(n!) / log(23) ).

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A127041 Maximal value of m such that 31^m <= n! : a(n) = floor( log(n!) / log(31) ).

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