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 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 (list; graph; refs; listen; history; text; internal format)
 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 ceil(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, 1, 200}]//Timing Code2[transparent]: Table[Count[Part[Sign[Table[Table[n!-n^j, {j, 1, 128}], {n, 1, 128}]], u], 1], {u, 1, 128}]//Timing (from Labos) 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

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Last modified January 21 19:08 EST 2019. Contains 319350 sequences. (Running on oeis4.)