A039960 For n >= 2, a(n) = largest value of k such that n^k is <= n! (a(0) = a(1) = 1 by convention).

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

Offset

0,5

Comments

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

Links

Danny Rorabaugh, Table of n, a(n) for n = 0..10000

Formula

a(n) = floor(log_n(n!)) for n > 1.

a(n) = A060151(n) - 1 for n > 1. - Henry Bottomley, Mar 08 2001

From Danny Rorabaugh, Apr 14 2015: (Start)

a(n) = log_n(A074182(n)) for n > 1.

a(n) = A074184 - 1 = log_n(A074181(n)) - 1 for n > 2. (End)

From Robert Israel, Apr 14 2015: (Start)

n*(1-1/log(n)) + 1 > log(n!)/log(n) > n*(1-1/log(n)) for n >= 7.

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)

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
(PARI) a(n)=if(n>3, lngamma(n+1)\log(n), 1) \\ Charles R Greathouse IV, Sep 02 2015

Crossrefs

Cf. A011776, A074181, A074182, A074184.

Sequence in context: A249243 A066530 A074183 * A105235 A073719 A105566

Adjacent sequences: A039957 A039958 A039959 * A039961 A039962 A039963

Keyword

nonn,easy

Author

Dan Bentley (bentini(AT)yahoo.com)

Extensions

Corrected and extended by Henry Bottomley, Mar 08 2001

Edited by N. J. A. Sloane, Sep 26 2008 at the suggestion of R. J. Mathar

Status

approved