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A375348 a(n) is the mode of the digits of n! not counting trailing zeros (using -1 if multimodal).

%I A375348 #21 Sep 15 2024 22:02:21
%S A375348 1,1,2,6,-1,-1,-1,-1,-1,8,8,9,0,2,-1,-1,8,-1,7,-1,-1,-1,7,8,3,1,6,8,
%T A375348 -1,-1,8,2,3,8,9,-1,9,-1,0,8,1,-1,-1,3,8,6,-1,1,7,2,6,-1,8,3,-1,5,4,2,
%U A375348 -1,8,4,0,2,6,-1,2,4,6,1,2,8,8,8,0,2,4,-1,8,2,1,5,7,4,-1,1,0
%N A375348 a(n) is the mode of the digits of n! not counting trailing zeros (using -1 if multimodal).
%C A375348 Inspired by A356758.
%C A375348 If we were to count trailing zeros, then would have a(n) = 0 for all n >= 34. Therefore we only consider the decimal digits of A004154(n).
%C A375348 Conjecture: excluding -1, as n -> oo, all digits occur equally often.
%e A375348 a(0) = a(1) = 1 because 0! = 1! = 1 and 1 is the only digit present;
%e A375348 a(4) = -1 since 4! = 24 and there are only two digits appearing with the same frequency, 2 and 4.
%e A375348 a(14) = -1 because 14! = 87178291200 and, not counting the two trailing 0's, there are two 1's, two 2's, two 7's, and two 8's.
%t A375348 a[n_] := If[Length[c=Commonest[IntegerDigits[n! / 10^IntegerExponent[n!]]]] > 1, -1, c[[1]]]; Array[a, 86, 0]
%o A375348 (Python)
%o A375348 from collections import Counter
%o A375348 from sympy import factorial
%o A375348 def A375348(n): return -1 if len(k:=Counter(str(factorial(n)).rstrip('0')).most_common(2)) > 1 and k[0][1]==k[1][1] else int(k[0][0]) # _Chai Wah Wu_, Sep 15 2024
%Y A375348 Cf. A004154, A027869, A031144, A034886, A061010, A137579, A137580, A356758.
%K A375348 base,easy,sign
%O A375348 0,3
%A A375348 _Stefano Spezia_ and _Robert G. Wilson v_, Aug 12 2024