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

%I A348467 #14 Oct 19 2021 23:48:24
%S A348467 1,1,1,1,3,6,6,7,11,20,25,33,32,41,60,72,80,106,104,132,140,150,173,
%T A348467 239,241,269,306,344,369,440,487,542,550,639,639,754,799,840,777,932,
%U A348467 1094,1032,1129,1203,1376,1440,1386,1681,1700,1737,1700,1948,1964,2099,2219
%N A348467 The number of distinct decimal representations of integers embedded as slices in the decimal representation of n!.
%F A348467 a(n) = A120004(n!). - _Michel Marcus_, Oct 19 2021
%e A348467 0:  1 // 1;
%e A348467 1:  1 // 1;
%e A348467 2:  1 // 2;
%e A348467 3:  1 // 6;
%e A348467 4:  3 // 2,4,24;
%e A348467 5:  6 // 0,1,2,12,20,120;
%e A348467 6:  6 // 0,2,7,20,72,720;
%e A348467 7:  7 // 0,4,5,40,50,504,5040;
%e A348467 8: 11 // 0,2,3,4,20,32,40,320,403,4032,40320;
%e A348467 9: 20 // 0,2,3,6,8,28,36,62,80,88,288,362,628,880,2880,3628,6288,36288,62880, 362880.
%t A348467 a[n_] := Length@ DeleteDuplicates[FromDigits /@ Rest@ Subsequences[ IntegerDigits[n!]]]; Array[a, 50, 0] (* _Amiram Eldar_, Oct 19 2021 *)
%o A348467 (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
%o A348467 a(n) = f(n!); \\ _Michel Marcus_, Oct 19 2021
%o A348467 (Python)
%o A348467 from math import factorial
%o A348467 def A348467(n):
%o A348467     s = str(factorial(n))
%o A348467     m = len(s)
%o A348467     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
%Y A348467 Cf. A120004, A219032, A000142.
%K A348467 nonn,base
%O A348467 0,5
%A A348467 _Peter Luschny_, Oct 19 2021