A079584 Number of ones in the binary expansion of n!.

1, 1, 1, 2, 2, 4, 4, 6, 6, 6, 11, 7, 12, 12, 12, 18, 18, 22, 23, 17, 22, 25, 28, 31, 29, 30, 35, 38, 42, 40, 48, 42, 42, 46, 51, 56, 51, 58, 59, 64, 63, 66, 64, 71, 74, 70, 77, 81, 89, 87, 89, 90, 88, 94, 87, 99, 103, 98, 101, 109, 113, 103, 113, 120, 120, 109, 123, 121, 130, 121

Offset

0,4

Links

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

Florian Luca, The number of non-zero digits of n!, Canad. Math. Bull. 45 (2002), pp. 115-118.

Carlo Sanna, On the sum of digits of the factorial, arXiv:1409.4912 [math.NT], 2014.

Carlo Sanna, On the sum of digits of the factorial, Journal of Number Theory 147 (2015), pp. 836-841.

Index entries for sequences related to factorial numbers

Formula

a(n) << n log n. - Charles R Greathouse IV, Mar 27 2013

a(n) = A000120(A000142(n)). - Michel Marcus, Sep 18 2014

Example

a(5) = 4 because 5! = 120 and 120_10 = 1111000_2, with 4 ones.

Maple

seq(convert(convert(n!, base, 2), `+`), n=0..1000); # Robert Israel, Sep 18 2014

Mathematica

Table[DigitCount[n!, 2, 1], {n, 70}] (* Harvey P. Dale, Jul 10 2012 *)

Prog

(PARI) for(n=1, 300, b=binary(n!); print1(sum(k=1, length(b), b[k])", "))
(PARI) a(n)=hammingweight(n!) \\ Charles R Greathouse IV, Mar 27 2013
(Python)
import math
def a(n):
    return bin(math.factorial(n))[2:].count("1") # Indranil Ghosh, Dec 23 2016

Crossrefs

Cf. A000120 (binary weight), A000142 (factorial), A004152 (sum of decimal digits).

Sequence in context: A070320 A124195 A220662 * A179291 A004079 A096494

Adjacent sequences: A079581 A079582 A079583 * A079585 A079586 A079587

Keyword

nonn,base

Author

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

Extensions

a(0)=1 prepended by Alois P. Heinz, Mar 07 2023

Status

approved