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%I #35 May 10 2015 09:54:16
%S 2,2,3,3,4,4,5,6,6,7,8,9,9,10,11,12,12,13,14,15,15,16,17,18,19,19,20,
%T 21,22,22,23,24,25,26,26,27,28,29,30,30,31,32,33,34,34,35,36,37,38,38,
%U 39,40,41,42,43,43,44,45,46,47,47,48,49,50,51,51,52,53,54
%N Exponents n that make k! < k^n < (k+1)! hold true for some integer k > 1, in increasing order by k, then n (if applicable).
%C This sequence consists of those positive integers that, when taken as exponents of some positive integer greater than 1, make the corresponding power of that other integer fall strictly between its factorial and the factorial of the next integer, as shown in the examples.
%C The sequence { floor(log_n((n+1)!)) | n>=2 } is a subsequence.
%C This sequence is nondecreasing. Indeed for k>1, k^n<(k+1)! implies n<=k, which implies ((k+1)/k)^(n-1) <= (1 + 1/k)^(k-1) = Sum_{i=0..k-1} binomial(k-1,i) (1/k)^i < Sum_{i=0..k-1} ((k-1)/k)^i < k, which implies (k+1)^(n-1)<k^n<(k+1)!. - _Danny Rorabaugh_, Apr 03 2015
%C From _Danny Rorabaugh_, Apr 15 2015: (Start)
%C This sequence is the same as A074184 for 6<=n<=10000.
%C For k > 2, k! < k^(ceiling(log_k(k!))) < (k+1)!.
%C The two sequences continue to be identical provided k^(1 + ceiling(log_k(k!))) > (k+1)! when k > 5.
%C This is equivalent to k^(2 - fractional_part(log_k(k!))) > k + 1, which can be approximated by fractional_part(1/2 - (k + sqrt(2*Pi))/log(k)) < 1 - 1/(k*log(k)) using Stirling's approximation.
%C Are either of the final inequalities true for all sufficiently large k?
%C (End)
%H Danny Rorabaugh, <a href="/A248868/b248868.txt">Table of n, a(n) for n = 1..10000</a>
%e 2! < 2^2 < 3! < 3^2 < 4! < 4^3 < 5! < 5^3 < 5^4 < 6! < 6^4 < 7! < 7^5 < 8! and so on; this sequence consists of the exponents.
%o (Sage)
%o [x for sublist in [[k for k in [0..ceil(log(factorial(n+1),base=n))] if (factorial(n)<n^k and n^k<factorial(n+1))] for n in [2..100]] for x in sublist] # _Tom Edgar_, Mar 04 2015
%Y Cf. A060151, A074181-A074184, A111683.
%K nonn
%O 1,1
%A _Juan Castaneda_, Mar 04 2015
%E More terms from _Tom Edgar_, Mar 04 2015
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