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%I #45 Apr 08 2024 06:54:49
%S 1,1,1,1,2,2,3,4,5,5,6,7,8,8,9,10,11,11,12,13,14,14,15,16,17,18,18,19,
%T 20,21,21,22,23,24,25,25,26,27,28,29,29,30,31,32,33,33,34,35,36,37,37,
%U 38,39,40,41,42,42,43,44,45,46,46,47,48,49,50,50,51,52,53,54,55,55,56,57
%N For n >= 2, a(n) = largest value of k such that n^k is <= n! (a(0) = a(1) = 1 by convention).
%C Seems to be slightly more than (but asymptotic to) number of nonprimes less than or equal to n.
%H Danny Rorabaugh, <a href="/A039960/b039960.txt">Table of n, a(n) for n = 0..10000</a>
%F a(n) = floor(log_n(n!)) for n > 1.
%F a(n) = A060151(n) - 1 for n > 1. - _Henry Bottomley_, Mar 08 2001
%F From _Danny Rorabaugh_, Apr 14 2015: (Start)
%F a(n) = log_n(A074182(n)) for n > 1.
%F a(n) = A074184 - 1 = log_n(A074181(n)) - 1 for n > 2. (End)
%F From _Robert Israel_, Apr 14 2015: (Start)
%F n*(1-1/log(n)) + 1 > log(n!)/log(n) > n*(1-1/log(n)) for n >= 7.
%F Thus a(n) is either floor(n*(1-1/log(n))) or ceiling(n*(1-1/log(n))) for n >= 7 (and in fact this is the case for n >= 3). (End)
%e a(7)=4 because 7! = 5040, 7^4 = 2401 but 7^5 = 16807.
%e a(6)=3 since 6^3.67195... = 720 = 6! and 6^3 <= 6! < 6^4, i.e., 216 <= 720 < 1296.
%t ds[x_, y_] :=y!-y^x; a[n_] :=Block[{m=1, s=ds[m, n]}, While[Sign[s]!=-1&&!Greater[m, 256], m++ ];m]; Table[a[n]-1, {n, 3, 200}]
%t (* or *)
%t Table[Count[Part[Sign[Table[Table[n!-n^j, {j, 1, 128}], {n, 1, 128}]], u], 1], {u, 1, 128}] (* _Labos Elemer_ *)
%t Join[{1,1},Table[Floor[Log[n,n!]],{n,2,80}]] (* _Harvey P. Dale_, Sep 24 2019 *)
%o (Sage) [1,1] + [floor(log(factorial(n))/log(n)) for n in range(2,75)] # _Danny Rorabaugh_, Apr 14 2015
%o (Magma) [1,1] cat [Floor(Log(Factorial(n))/Log(n)): n in [2..80]]; // _Vincenzo Librandi_, Apr 15 2015
%o (PARI) a(n)=if(n>3,lngamma(n+1)\log(n),1) \\ _Charles R Greathouse IV_, Sep 02 2015
%Y Cf. A011776, A074181, A074182, A074184.
%K nonn,easy
%O 0,5
%A Dan Bentley (bentini(AT)yahoo.com)
%E Corrected and extended by _Henry Bottomley_, Mar 08 2001
%E Edited by _N. J. A. Sloane_, Sep 26 2008 at the suggestion of _R. J. Mathar_
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