

A039960


For n >= 2, a(n) = largest value of k such that n^k is <= n! (a(0) = a(1) = 1 by convention).


7



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, 20, 21, 21, 22, 23, 24, 25, 25, 26, 27, 28, 29, 29, 30, 31, 32, 33, 33, 34, 35, 36, 37, 37, 38, 39, 40, 41, 42, 42, 43, 44, 45, 46, 46, 47, 48, 49, 50, 50, 51, 52, 53, 54, 55, 55, 56, 57
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OFFSET

0,5


COMMENTS

Seems to be slightly more than (but asymptotic to) number of nonprimes less than or equal to n.


LINKS



FORMULA

a(n) = floor(log_n(n!)) for n > 1.
a(n) = log_n(A074182(n)) for n > 1.
n*(11/log(n)) + 1 > log(n!)/log(n) > n*(11/log(n)) for n >= 7.
Thus a(n) is either floor(n*(11/log(n))) or ceiling(n*(11/log(n))) for n >= 7 (and in fact this is the case for n >= 3). (End)


EXAMPLE

a(7)=4 because 7! = 5040, 7^4 = 2401 but 7^5 = 16807.
a(6)=3 since 6^3.67195... = 720 = 6! and 6^3 <= 6! < 6^4, i.e., 216 <= 720 < 1296.


MATHEMATICA

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}]
(* or *)
Table[Count[Part[Sign[Table[Table[n!n^j, {j, 1, 128}], {n, 1, 128}]], u], 1], {u, 1, 128}] (* Labos Elemer *)
Join[{1, 1}, Table[Floor[Log[n, n!]], {n, 2, 80}]] (* Harvey P. Dale, Sep 24 2019 *)


PROG

(Sage) [1, 1] + [floor(log(factorial(n))/log(n)) for n in range(2, 75)] # Danny Rorabaugh, Apr 14 2015
(Magma) [1, 1] cat [Floor(Log(Factorial(n))/Log(n)): n in [2..80]]; // Vincenzo Librandi, Apr 15 2015


CROSSREFS



KEYWORD

nonn,easy


AUTHOR

Dan Bentley (bentini(AT)yahoo.com)


EXTENSIONS



STATUS

approved



