A348467 The number of distinct decimal representations of integers embedded as slices in the decimal representation of n!.

1, 1, 1, 1, 3, 6, 6, 7, 11, 20, 25, 33, 32, 41, 60, 72, 80, 106, 104, 132, 140, 150, 173, 239, 241, 269, 306, 344, 369, 440, 487, 542, 550, 639, 639, 754, 799, 840, 777, 932, 1094, 1032, 1129, 1203, 1376, 1440, 1386, 1681, 1700, 1737, 1700, 1948, 1964, 2099, 2219

Offset: 0

Author

Peter Luschny, Oct 19 2021

Keywords

• nonn
• base

Examples

			0:  1 // 1;
1:  1 // 1;
2:  1 // 2;
3:  1 // 6;
4:  3 // 2,4,24;
5:  6 // 0,1,2,12,20,120;
6:  6 // 0,2,7,20,72,720;
7:  7 // 0,4,5,40,50,504,5040;
8: 11 // 0,2,3,4,20,32,40,320,403,4032,40320;
9: 20 // 0,2,3,6,8,28,36,62,80,88,288,362,628,880,2880,3628,6288,36288,62880, 362880.
		

Crossrefs

Cf. A120004, A219032, A000142.

Programs

Mathematica

a[n_] := Length@ DeleteDuplicates[FromDigits /@ Rest@ Subsequences[ IntegerDigits[n!]]]; Array[a, 50, 0] (* Amiram Eldar, Oct 19 2021 *)

PARI

f(n) = if (n==0, return (1)); my(d=digits(n), list=List()); for (k=1, #d, for (j=1, #d-k+1, my(dk=vector(j, i, d[k+i-1])); listput(list, fromdigits(dk)););); #Set(list); \\ A120004
a(n) = f(n!); \\ Michel Marcus, Oct 19 2021

Python

from math import factorial
def A348467(n):
    s = str(factorial(n))
    m = len(s)
    return len(set(int(s[i:j]) for i in range(m) for j in range(i+1,m+1))) # Chai Wah Wu, Oct 19 2021

Formula

a(n) = A120004(n!). - Michel Marcus, Oct 19 2021