A027869 Number of 0's in n!.

0, 0, 0, 0, 0, 1, 1, 2, 2, 1, 2, 2, 4, 4, 2, 4, 4, 4, 5, 6, 7, 7, 8, 5, 6, 9, 8, 9, 10, 7, 9, 7, 10, 8, 11, 9, 10, 12, 16, 12, 9, 15, 13, 13, 12, 13, 16, 11, 14, 14, 19, 18, 18, 17, 18, 18, 17, 20, 17, 19, 19, 26, 20, 21, 20, 20, 23, 22, 25, 21, 20, 25, 23, 35

Offset

0,8

Links

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

Index entries for sequences related to factorial numbers.

Formula

a(n) = A034886(n) - (A079680(n) + A079714(n) + A079684(n) + A079688(n) + A079690(n) + A079691(n) + A079692(n) + A079693(n) + A079694(n)). - Reinhard Zumkeller, Jan 27 2008

A027868(n) <= a(n). - Reinhard Zumkeller, Jan 27 2008

Conjecture: a(n) ~ (9*A027868(n) + A034886(n))/10. This formula is based on the assumption that the digits other than trailing zeros are uniformly randomly distributed. - Nicolas Bělohoubek, Jan 11 2022

Mathematica

Table[Count[IntegerDigits[n!], 0], {n, 0, 100}] (* T. D. Noe, Apr 10 2012 *)
DigitCount[Range[0, 80]!, 10, 0] (* Harvey P. Dale, Jul 08 2020 *)

Prog

(PARI) a(n)=my(d=digits(n!)); sum(i=1, #d, d[i]==0) \\ Charles R Greathouse IV, Jul 06 2017
(Python)
from math import factorial
def a(n): return str(factorial(n)).count('0')
print([a(n) for n in range(74)]) # Michael S. Branicky, Jan 11 2022

Crossrefs

Cf. A000142, A137579, A137580, A034886.

Sequence in context: A334590 A115034 A253193 * A156748 A351455 A075445

Adjacent sequences: A027866 A027867 A027868 * A027870 A027871 A027872

Keyword

nonn,base

Author

N. J. A. Sloane

Status

approved