A137579 -id:A137579 - cp's OEIS frontend

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

N. J. A. Sloane

Alois P. Heinz, Table of n, a(n) for n = 0..10000 (first 1001 terms from T. D. Noe)
Index entries for sequences related to factorial numbers.

Cf. A000142, A137579, A137580, A034886.

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 *)

a(n)=my(d=digits(n!)); sum(i=1,#d,d[i]==0) \\ Charles R Greathouse IV, Jul 06 2017

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

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

A137580 Number of distinct digits in decimal representation of n!.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 3, 3, 4, 5, 5, 6, 6, 5, 6, 7, 5, 9, 8, 8, 9, 7, 7, 10, 9, 8, 9, 10, 8, 9, 9, 10, 9, 10, 10, 10, 10, 10, 9, 10, 10, 9, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10

Offset: 0

Reinhard Zumkeller, Jan 27 2008

			n=12: 12! = 479001600 => a(12) = #{0,1,4,6,7,9} = 6.

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
Index entries for sequences related to factorial numbers.

Cf. A137577, A137578.

Cf. A034886.

import Data.List (nub, sort)
a137580 = length . nub . show . a000142
-- Reinhard Zumkeller, Apr 08 2012

Map[Length[Union[IntegerDigits[#]]] &, Table[n!, {n, 0, 79}]] (* Geoffrey Critzer, May 25 2013 *)

A137580(n)=#Set(digits(n!)) \\ M. F. Hasler, May 04 2015

a(n) = A043537(A000142(n)).
a(n) < 10 iff A137579(n) = 0.
a(A182049(n)) < 10. - Reinhard Zumkeller, Apr 08 2012

A079692 Number of 7's in n!.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 2, 2, 1, 1, 3, 0, 1, 2, 6, 2, 1, 0, 0, 1, 1, 3, 0, 4, 1, 1, 2, 2, 4, 3, 4, 4, 4, 3, 3, 4, 4, 4, 1, 2, 8, 5, 5, 3, 8, 5, 7, 4, 9, 4, 4, 7, 7, 6, 8, 8, 3, 9, 8, 6, 8, 8, 9, 10, 12, 7, 7, 9, 9, 7, 10, 10, 9, 14, 11, 12, 9, 7, 13, 17, 2, 11, 12, 19, 15, 12, 10, 15, 15, 16, 19, 7, 7, 12

Offset: 0

Cino Hilliard, Jan 31 2003

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Index entries for sequences related to factorial numbers.

Cf. A079714.
Cf. A000142, A137579, A137580.
Cf. A316868.

a:= n-> numboccur(7, convert(n!, base, 10)):
seq(a(n), n=0..101);  # Alois P. Heinz, Apr 26 2021

a(n) = #select(x->(x==7), digits(n!)); \\ Michel Marcus, Apr 26 2021

a(n) = A034886(n) - (A027869(n) + A079680(n) + A079714(n) + A079684(n) + A079688(n) + A079690(n) + A079691(n) + A079693(n) + A079694(n)). - Reinhard Zumkeller, Jan 27 2008

a(78)-a(79) corrected by Georg Fischer, Apr 26 2021

A079680 Number of 1's in n!.

Original entry on oeis.org

1, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 2, 1, 0, 0, 0, 3, 1, 3, 2, 1, 1, 4, 3, 3, 4, 4, 3, 2, 5, 7, 2, 4, 4, 4, 7, 3, 6, 6, 6, 4, 6, 7, 4, 10, 4, 5, 7, 10, 3, 2, 5, 5, 4, 7, 10, 6, 8, 3, 8, 9, 12, 5, 5, 8, 11, 9, 7, 6, 6, 16, 13, 9, 7, 7, 11, 15, 8, 9, 13, 10, 15, 13, 8, 14, 14, 12, 11, 12, 13, 16, 11

Offset: 0

Cino Hilliard, Jan 31 2003

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Index entries for sequences related to factorial numbers.

Cf. A000142, A137579, A137580.

a(n) = A034886(n) - (A027869(n) + A079714(n) + A079684(n) + A079688(n) + A079690(n) + A079691(n) + A079692(n) + A079693(n) + A079694(n)). - Reinhard Zumkeller, Jan 27 2008

A079684 Number of 3's in n!.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 2, 0, 1, 2, 1, 1, 0, 0, 1, 5, 3, 2, 1, 3, 3, 2, 1, 7, 2, 3, 7, 4, 5, 1, 5, 3, 5, 3, 9, 3, 5, 1, 5, 7, 6, 6, 6, 4, 9, 8, 5, 3, 4, 5, 8, 8, 4, 8, 5, 9, 7, 7, 6, 9, 10, 5, 7, 8, 6, 10, 7, 11, 7, 9, 10, 8, 8, 15, 10, 13, 8, 10, 13, 8, 12, 10, 6, 18, 12, 12, 15, 9, 12

Offset: 0

Cino Hilliard, Jan 31 2003

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Index entries for sequences related to factorial numbers.

Cf. A079714.

Cf. A000142, A137579, A137580.

DigitCount[Range[0,100]!,10,3] (* Harvey P. Dale, Jul 04 2014 *)

a(n) = A034886(n) - (A027869(n) + A079680(n) + A079714(n) + A079688(n) + A079690(n) + A079691(n) + A079692(n) + A079693(n) + A079694(n)). - Reinhard Zumkeller, Jan 27 2008

Corrected by Jason Earls, Jul 06 2003

A079688 Number of 4's in n!.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 1, 2, 2, 3, 1, 2, 4, 2, 3, 2, 4, 3, 1, 1, 0, 3, 4, 3, 3, 3, 3, 3, 4, 3, 6, 2, 5, 4, 3, 3, 4, 3, 7, 3, 5, 7, 6, 6, 13, 8, 8, 7, 10, 4, 4, 8, 10, 8, 16, 10, 7, 13, 6, 5, 10, 7, 7, 13, 11, 11, 10, 4, 13, 13, 16, 10, 8, 15, 14, 10, 18, 6, 13, 12, 17, 12

Offset: 0

Cino Hilliard, Jan 31 2003

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Index entries for sequences related to factorial numbers.

Cf. A079714.

Cf. A000142, A137579, A137580.

DigitCount[#,10,4]&/@(Range[0,100]!) (* Harvey P. Dale, Jul 30 2015 *)

a(n) = A034886(n) - (A027869(n) + A079680(n) + A079714(n) + A079684(n) + A079690(n) + A079691(n) + A079692(n) + A079693(n) + A079694(n)). - Reinhard Zumkeller, Jan 27 2008

A079690 Number of 5's in n!.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 1, 1, 0, 1, 0, 2, 0, 3, 3, 2, 2, 2, 4, 3, 1, 2, 2, 2, 2, 5, 1, 2, 5, 6, 5, 7, 5, 5, 8, 5, 6, 5, 2, 4, 7, 3, 3, 11, 5, 6, 5, 5, 3, 7, 6, 4, 7, 10, 3, 7, 8, 5, 10, 7, 3, 7, 13, 9, 10, 6, 9, 7, 14, 13, 1, 12, 8, 13, 13, 11, 10, 10, 12, 17, 17, 14, 15, 12

Offset: 0

Cino Hilliard, Jan 31 2003

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Index entries for sequences related to factorial numbers.

Cf. A027869, A079714.

Cf. A000142, A137579, A137580.

DigitCount[#,10,5]&/@(Range[0,100]!) (* Harvey P. Dale, Sep 17 2016 *)

a(n) = A034886(n) - (A027869(n) + A079680(n) + A079714(n) + A079684(n) + A079688(n) + A079691(n) + A079692(n) + A079693(n) + A079694(n)). - Reinhard Zumkeller, Jan 27 2008

A079691 Number of 6's in n!.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 0, 2, 0, 2, 1, 1, 2, 0, 2, 3, 2, 0, 4, 3, 2, 3, 3, 2, 5, 3, 4, 7, 2, 3, 5, 2, 3, 6, 5, 6, 5, 8, 4, 7, 6, 6, 9, 5, 8, 7, 3, 9, 6, 7, 4, 6, 8, 6, 6, 11, 6, 8, 8, 11, 6, 4, 11, 11, 10, 6, 5, 9, 8, 9, 8, 11, 10, 8, 11, 12, 13, 9, 11, 7, 12, 15, 15, 17, 8, 11, 16, 11

Offset: 0

Cino Hilliard, Jan 31 2003

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Index entries for sequences related to factorial numbers.

Cf. A079714.

Cf. A000142, A137579, A137580.

Table[DigitCount[n!,10,6],{n,0,100}] (* Harvey P. Dale, Aug 08 2023 *)

a(n) = A034886(n) - (A027869(n) + A079680(n) + A079714(n) + A079684(n) + A079688(n) + A079690(n) + A079692(n) + A079693(n) + A079694(n)). - Reinhard Zumkeller, Jan 27 2008

A079693 Number of 8's in n!.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 1, 0, 1, 2, 1, 4, 2, 1, 2, 1, 0, 1, 4, 1, 2, 1, 6, 4, 2, 5, 6, 2, 8, 2, 1, 3, 2, 0, 7, 4, 2, 4, 2, 9, 3, 7, 4, 4, 7, 5, 5, 9, 8, 9, 4, 11, 7, 10, 9, 4, 11, 7, 7, 12, 7, 9, 9, 7, 8, 14, 18, 15, 9, 9, 10, 8, 18, 12, 14, 13, 8, 10, 8, 12, 5, 8, 8, 18, 10, 14, 9, 11, 12, 16

Offset: 0

Cino Hilliard, Jan 31 2003

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Index entries for sequences related to factorial numbers.

Cf. A079714.

Cf. A000142, A137579, A137580.

Table[DigitCount[n!,10,8],{n,0,100}] (* Harvey P. Dale, May 06 2016 *)

a(n) = A034886(n) - (A027869(n) + A079680(n) + A079714(n) + A079684(n) + A079688(n) + A079690(n) + A079691(n) + A079692(n) + A079694(n)). - Reinhard Zumkeller, Jan 27 2008

A079694 Number of 9's in n!.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 1, 0, 1, 0, 2, 1, 0, 0, 1, 3, 0, 1, 2, 2, 1, 1, 0, 4, 2, 1, 2, 2, 5, 3, 7, 4, 1, 5, 5, 0, 4, 2, 2, 4, 6, 7, 3, 2, 2, 3, 3, 6, 4, 6, 6, 5, 6, 8, 6, 7, 6, 7, 5, 6, 6, 8, 8, 7, 12, 5, 7, 5, 7, 10, 12, 7, 6, 9, 5, 12, 13, 12, 10, 9, 9, 10, 13, 18, 14, 12, 7, 7, 7, 15, 20, 16

Offset: 0

Cino Hilliard, Jan 31 2003

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Index entries for sequences related to factorial numbers.

Cf. A027869, A079714.

Cf. A000142, A137579, A137580.

DigitCount[#,10,9]&/@(Range[0,100]!) (* Harvey P. Dale, Dec 12 2013 *)

a(n) = A034886(n) - (A027869(n) + A079680(n) + A079714(n) + A079684(n) + A079688(n) + A079690(n) + A079691(n) + A079692(n) + A079693(n)). - Reinhard Zumkeller, Jan 27 2008

cp's OEIS Frontend

A027869 Number of 0's in n!.

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A137580 Number of distinct digits in decimal representation of n!.

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A079692 Number of 7's in n!.

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A079680 Number of 1's in n!.

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A079684 Number of 3's in n!.

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A079688 Number of 4's in n!.

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A079690 Number of 5's in n!.

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A079691 Number of 6's in n!.

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A079693 Number of 8's in n!.