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A216377 The most significant digit in base n representation of n!.

%I A216377 #27 Jan 01 2025 14:40:13
%S A216377 1,2,1,4,3,2,1,6,3,2,1,7,4,2,1,10,5,2,1,15,8,4,2,1,13,6,3,1,25,12,6,3,
%T A216377 1,25,12,6,3,1,28,13,6,3,1,33,16,7,3,1,41,20,9,4,2,1,26,12,6,2,1,38,
%U A216377 18,8,3,1,57,27,12,5,2,1,43,20,9,4,2,72,33,15,7,3,1
%N A216377 The most significant digit in base n representation of n!.
%C A216377 a(n) < n, by definition.
%C A216377 Numbers n such that a(n)=1: 2, 4, 8, 12, 16, 20, 25, 29, 34, 39, 44, 49, 55, 60, 65, 71, 82, 88, 94, 105, 111, 117, 123, 136, ... (see A221707).
%C A216377 Numbers n such that a(n) > a(k) for k < n: 2, 3, 5, 9, 13, 17, 21, 30, 40, 45, 50, 66, 77, 100, 118, 124, 130, 155, 161, 226, 246, 273, 371, 378, 385, 421, 450, 472, 509, 584, 599, 637, 660, 683, 745, 784, 855, 983, 991, 999, ... (see A221708).
%H A216377 Alois P. Heinz, <a href="/A216377/b216377.txt">Table of n, a(n) for n = 2..10000</a>
%F A216377 a(n) = modlg(n!, n), where modlg is the function defined in A215894: modlg(A,B) = floor(A / B^floor(logB(A))), logB is the logarithm base B.
%p A216377 a:= n-> iquo(n!, n^ilog[n](n!)):
%p A216377 seq(a(n), n=2..100);  # _Alois P. Heinz_, Sep 06 2012
%t A216377 Table[IntegerDigits[n!, n][[1]], {n, 2, 100}] (* _T. D. Noe_, Sep 06 2012 *)
%o A216377 (Python)
%o A216377 import math
%o A216377 def modlg(a,b):
%o A216377     return a // b**int(math.log(a,b))
%o A216377 for n in range(2,88):
%o A216377     print(modlg(math.factorial(n), n), end=', ')
%Y A216377 Cf. A216021, A215894, A221707, A221708.
%Y A216377 Cf. also to scatter plots of A265891 and A265894.
%K A216377 nonn,base,look
%O A216377 2,2
%A A216377 _Alex Ratushnyak_, Sep 06 2012