A316868 -id:A316868 - cp's OEIS frontend

A079692 Number of 7's in n!.

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

A100910 Table of number of occurrences in n of each decimal digit from 0 to 9.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0

Offset: 0

Rick L. Shepherd, Nov 21 2004

Each row of this table has length 10 and corresponds to one term of A100909. n = 0 is normally represented as the single digit 0, so the first row here is 1, 0, 0, 0, 0, 0, 0, 0, 0, 0.

Robert Israel, Table of n, a(n) for n = 0..10009 (rows 0 to 1000, flattened)

Cf. A100909 (similar but each row of A100910 provides one A100909 term).
Cf. A055642 (row sums), A055641 (column 0), A268643 (column 1), A316863 (column 2), A316864 (column 3), A316865 (column 4), A316866 (column 5), A316867 (column 6), A316868 (column 7), A316869 (column 8), A102683 (column 9).
Cf. A059995, A010879.

seq(seq(numboccur(k, convert(n,base,10)),k=0..9),n=0..100); # Robert Israel, Jul 08 2016

A100910row[n_] := RotateRight[DigitCount[n]];
Array[A100910row, 10, 0] (* Paolo Xausa, Jul 16 2025 *)

T(n, k) = #select(x->x==k, digits(n))+!(n+k); \\ Jinyuan Wang, Mar 01 2020

From Robert Israel, Jul 08 2016: (Start)
a(n,k) = a(A059995(n),k) + (1 if A010879(n)=k, otherwise 0).
G.f. g(x,y) satisfies g(x,y) = ((1-x^10)/(1-x))*g(x^10,y) + (x^10-x)/(1-x) + x^10/(1-x^10) + x*y*(1-x^9*y^9)/((1-x^10)*(1-x*y)). (End)

A316863 Number of times 2 appears in the decimal expansion of n.

Original entry on oeis.org

0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0

Offset: 0

Robert G. Wilson v, Jul 15 2018

			a(0) = 0 since the decimal representation of 0 does not contain the digit 2.
a(2) = 1 since 2 appears once in the decimal expansion of 2.
a(22) = 2 since 2 appears twice in the decimal expansion of 22.

Robert Israel, Table of n, a(n) for n = 0..10000

Cf. A043497, A043498, A043499, A043500.

Cf. A055641, A268643, A316864, A316865, A316866, A316867, A316868, A316869, A102683.

f:= n -> numboccur(2,convert(n,base,10)):
map(f, [$0..200]); # Robert Israel, Apr 21 2020

Mathematica
```
Array[ DigitCount[#, 10, 2] &, 105, 0]
```

a(n) = #select(x->x==2, digits(n)); \\ Michel Marcus, Jul 20 2018

G.f.: (1-x)^(-1)*Sum_{k>=0} (x^(2*10^k)-x^(3*10^k))/(1-x^(10^(k+1))). - Robert Israel, Apr 21 2020

A316864 Number of times 3 appears in decimal expansion of n.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0

Offset: 0

Robert G. Wilson v, Jul 15 2018

			a(0) = 0 since the decimal representation of 0 does not contain the digit 3.
a(3) = 1 since 3 appears once in the decimal expansion of 3.

Robert Israel, Table of n, a(n) for n = 0..10000

Cf. A043501, A043502, A043503, A043504.

Cf. A055641, A268643, A316864, A316865, A316866, A316867, A316868, A316869, A102683.

f:= proc(n) option remember;
procname(floor(n/10)) + `if`(n mod 10 = 3, 1, 0)
end proc:
for i from 0 to 9 do f(i):= `if`(i=3,1,0) od:
map(f, [$0..100]); # Robert Israel, Dec 10 2019

Mathematica
```
Array[ DigitCount[#, 10, 3] &, 105, 0]
```

a(n) = #select(x->x==3, digits(n)); \\ Michel Marcus, Jul 20 2018

From Robert Israel, Dec 10 2019: (Start)
a(10*n+3) = a(n)+1, a(10*n+i)=a(i) for i = 0,1,2,4..9.
G.f. g(z) satisfies g(z) = z^3/(1-z^10) + ((1-z^10)/(1-z))*g(z^10). (End)

A316869 Number of times 8 appears in decimal expansion of n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0

Offset: 0

Robert G. Wilson v, Jul 15 2018

			a(0) = 0 since the decimal representation of 0 does not contain the digit 8.
a(8) = 1 since 8 appears once in the decimal expansion of 8.

Cf. A043521, A043522, A043523, A043524.

Cf. A055641, A268643, A316863, A316864, A316865, A316866, A316867, A316868, A102683.

Mathematica
```
Array[ DigitCount[#, 10, 8] &, 105, 0]
```

a(n) = #select(x->x==8, digits(n)); \\ Michel Marcus, Jul 20 2018

A316865 Number of times 4 appears in decimal expansion of n.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1

Offset: 0

Robert G. Wilson v, Jul 15 2018

			a(0) = 0 since the decimal representation of 0 does not contain the digit 4.
a(4) = 1 since 4 appears once in the decimal expansion of 4.

Cf. A043505, A043506, A043507, A043508.

Cf. A055641, A268643, A316863, A316864, A316866, A316867, A316868, A316869, A102683.

Mathematica
```
Array[ DigitCount[#, 10, 4] &, 105, 0]
```

a(n) = #select(x->x==4, digits(n)); \\ Michel Marcus, Jul 20 2018

A316866 Number of times 5 appears in decimal expansion of n.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Robert G. Wilson v, Jul 15 2018

			a(0) = 0 since the decimal representation of 0 does not contain the digit 5.
a(5) = 1 since 5 appears once in the decimal expansion of 5.

Cf. A043509, A043510, A043511, A043512.

Cf. A055641, A268643, A316863, A316864, A316865, A316867, A316868, A316869, A102683.

Mathematica
```
Array[ DigitCount[#, 10, 5] &, 105, 0]
```

a(n) = #select(x->x==5, digits(n)); \\ Michel Marcus, Jul 20 2018

A316867 Number of times 6 appears in decimal expansion of n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Robert G. Wilson v, Jul 15 2018

			a(0) = 0 since the decimal representation of 0 does not contain the digit 6.
a(6) = 1 since 6 appears once in the decimal expansion of 6.

Cf. A043513, A043514, A043515, A043516.

Cf. A055641, A268643, A316863, A316864, A316865, A316866, A316868, A316869, A102683.

Mathematica
```
Array[ DigitCount[#, 10, 6] &, 105, 0]
```

a(n) = #select(x->x==6, digits(n)); \\ Michel Marcus, Jul 20 2018

A342115 Lexicographically latest sequence of distinct nonnegative integers such that the multisets of frequencies of digits in the decimal representations of n and of a(n) are the same.

Original entry on oeis.org

0, 9, 8, 7, 6, 5, 4, 3, 2, 1, 98, 99, 97, 96, 95, 94, 93, 92, 91, 90, 89, 87, 88, 86, 85, 84, 83, 82, 81, 80, 79, 78, 76, 77, 75, 74, 73, 72, 71, 70, 69, 68, 67, 65, 66, 64, 63, 62, 61, 60, 59, 58, 57, 56, 54, 55, 53, 52, 51, 50, 49, 48, 47, 46, 45, 43, 44, 42

Offset: 0

Rémy Sigrist, Feb 28 2021

We ignore leading zeros (hence a(0) = 0).

This sequence is a self-inverse permutation of the nonnegative integers.

			Consider the set of numbers T with two distinct digits, say u and v, such that u appears once and v appears twice:
- the least elements of T are: 100, 101, 110, 112,
- the greatest elements of T are: 995, 996, 997, 998,
- so a(100) = 998, a(101) = 997, a(110) = 996, a(112) = 995.

Rémy Sigrist, Table of n, a(n) for n = 0..9999
Rémy Sigrist, PARI program for A342115
Index entries for sequences that are permutations of the natural numbers

See A342102 for similar sequences.

Cf. A055641, A268643, A316863, A316864, A316865, A316866, A316867, A316868, A316869, A102683.

PARI
```
See Links section.
```

a(d * (10^k-1)/9) = (10-d) * (10^k-1)/9 for any k >= 0 and d = 1..9.

a(n) < 10^k for any n < 10^k.

cp's OEIS Frontend

A079692 Number of 7's in n!.

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A100910 Table of number of occurrences in n of each decimal digit from 0 to 9.

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A316863 Number of times 2 appears in the decimal expansion of n.

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A316864 Number of times 3 appears in decimal expansion of n.

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A316869 Number of times 8 appears in decimal expansion of n.

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A316865 Number of times 4 appears in decimal expansion of n.

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A316866 Number of times 5 appears in decimal expansion of n.

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A316867 Number of times 6 appears in decimal expansion of n.

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