A072831 -id:A072831 - cp's OEIS frontend

A003070 a(n) = ceiling(log_2 n!).

0, 0, 1, 3, 5, 7, 10, 13, 16, 19, 22, 26, 29, 33, 37, 41, 45, 49, 53, 57, 62, 66, 70, 75, 80, 84, 89, 94, 98, 103, 108, 113, 118, 123, 128, 133, 139, 144, 149, 154, 160, 165, 170, 176, 181, 187, 192, 198, 203, 209, 215, 220, 226, 232, 238, 243, 249, 255, 261, 267

Offset: 0

N. J. A. Sloane

nonn

a(n) is a lower bound for the minimum number of comparisons needed to sort n elements using a comparison sort (A036604). - Alex Costea, Mar 23 2019

D. E. Knuth, Art of Computer Programming, Vol. 3, Sect. 5.3.1.
E. Reingold, J. Nievergelt and N. Deo, Combinatorial Algorithms, Prentice-Hall, 1977, section 7.4.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Tianxing Tao, On optimal arrangement of 12 points, pp. 229-234 in Combinatorics, Computing and Complexity, ed. D. Du and G. Hu, Kluwer, 1989.

Alois P. Heinz, Table of n, a(n) for n = 0..10000
Index entries for sequences related to sorting

Cf. A036604. Essentially the same as A072831.

[Ceiling(Log(2,Factorial(n))) : n in [0..70]]; // G. C. Greubel, Nov 03 2022

Array[Ceiling@ Log2[#!] &, 60, 0] (* Michael De Vlieger, Mar 27 2019 *)

[ceil(log(factorial(n),2)) for n in range(71)] # G. C. Greubel, Nov 03 2022

A213857 Least m such that n! <= 3^m.

Original entry on oeis.org

1, 1, 2, 3, 5, 6, 8, 10, 12, 14, 16, 19, 21, 23, 26, 28, 31, 34, 36, 39, 42, 45, 47, 50, 53, 56, 59, 62, 65, 68, 72, 75, 78, 81, 84, 88, 91, 94, 98, 101, 104, 108, 111, 115, 118, 122, 125, 129, 132, 136, 139, 143, 146, 150, 154, 157, 161, 165, 168, 172, 176

Offset: 1

Clark Kimberling, Jul 17 2012

Also the number of digits in the ternary representation of n!. - Martin Renner, Jan 03 2022

			a(8) = 10 because 3^9 < 8! <= 3^10.

Clark Kimberling, Table of n, a(n) for n = 1..1000

Cf. A003070, A072831, A213854.

seq(nops(convert(n!,base,3)),n=1..100); # Martin Renner, Jan 03 2022

Table[m=1; While[n!>3^m, m++]; m, {n,1,100}]

A093710 Numbers k such that in their binary representation all numbers from 1 to k are contained in k!.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 8, 14, 23, 26, 28, 30, 33, 34, 35, 39, 42, 43, 51, 53, 58, 61, 62, 63, 64, 66, 68, 70, 73, 77, 80, 83, 93, 94, 106, 108, 111, 114, 115, 116, 126, 131, 132, 133, 134, 136, 137, 147, 149, 153, 155, 156, 169, 172, 175, 180, 185, 187, 191, 195, 206

Offset: 1

Reinhard Zumkeller, Apr 11 2004

			6 is in the sequence because 6! = 1011010000_2 which contains 4 = 100_2, 5 = 101_2 and 6 = 110_2 as a substring in the binary expansion. As it contains 4, 5 and 6 in binary it contains the binary expansion of every smaller number than 4 in its binary expansion. - _David A. Corneth_, Aug 05 2025

David A. Corneth, Table of n, a(n) for n = 1..10000 (first 3773 terms from Amiram Eldar)
David A. Corneth, Plot of a(n+1)-a(n), n = 1..9999
David A. Corneth, plot of a(n)/n, n = 1..10000

Complement of A093711.

Cf. A000142, A007088, A036603, A072831, A092601.

A092601(a(n)) = a(n).

A078565 Number of zeros in the binary expansion of n!.

Original entry on oeis.org

0, 0, 1, 1, 3, 3, 6, 7, 10, 13, 11, 19, 17, 21, 25, 23, 27, 27, 30, 40, 40, 41, 42, 44, 51, 54, 54, 56, 56, 63, 60, 71, 76, 77, 77, 77, 88, 86, 90, 90, 97, 99, 106, 105, 107, 117, 115, 117, 114, 122, 126, 130, 138, 138, 151, 144, 146, 157, 160, 158, 160, 176

Offset: 0

Jose R. Brox (tautocrona(AT)terra.es), Jan 26 2003

			a(4) = 3 because 4! = 24_10 = 11000_2 and it has 3 zero digits.

T. D. Noe, Table of n, a(n) for n = 0..1000

Cf. A072831, A079584.

Table[Count[IntegerDigits[n!, 2], 0], {n, 0, 100}] (* T. D. Noe, Apr 10 2012 *)
DigitCount[Range[0,70]!,2,0] (* Harvey P. Dale, Mar 11 2019 *)

for(n=1,300,b=binary(n!); print1(sum(k=1,length(b),if(b[k],0,1))","))

import math
def a(n):
    return bin(math.factorial(n))[2:].count("0")
# Indranil Ghosh, Dec 23 2016

a(n) = A072831(n) - A079584(n). - Michel Marcus, Dec 23 2016

A152168 Number of binary digits in (n!)!.

Original entry on oeis.org

1, 1, 2, 10, 80, 661, 5802, 54725, 558704, 6178565, 73840164, 950331113, 13121175977, 193618002604, 3042570732326, 50747501675076, 895651186352884, 16679929313440954, 326936145826028780, 6728526339596831313, 145085354333183129464, 3271200076443827203823

Offset: 0

Jon E. Schoenfield, Nov 27 2008

			(3!)! = 6! = 720 has ten binary digits (1011010000), so a(3) = 10.

Jon E. Schoenfield, Table of n, a(n) for n = 0..30
Index entries for sequences related to factorial numbers

Cf. A000197, A072831.

a(n) = length(binary(n!!)) \\ Michel Marcus, Jun 02 2013

cp's OEIS Frontend

A003070 a(n) = ceiling(log_2 n!).

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A213857 Least m such that n! <= 3^m.

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A093710 Numbers k such that in their binary representation all numbers from 1 to k are contained in k!.

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A078565 Number of zeros in the binary expansion of n!.

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A152168 Number of binary digits in (n!)!.

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