A038024 Number of k's such that A002034(k) = n.

1, 1, 2, 4, 8, 14, 30, 36, 64, 110, 270, 252, 792, 1008, 1440, 1344, 5376, 3936, 14688, 11664, 19760, 35200, 96000, 50880, 97152, 192192, 145152, 239904, 917280, 498240, 2332800, 864000, 2334720, 4300800, 4257792, 3172608

Offset: 1

Author

Christian G. Bower

Keywords

• nonn

Links

• Paul Erdős, S. W. Graham, Alexsandr Ivić, and Carl Pomerance, On the number of divisors of n!, Analytic Number Theory, Proceedings of a Conference in Honor of Heini Halberstam, ed. by B. C. Berndt, H. G. Diamond, A. J. Hildebrand, Birkhauser 1996, pp. 337-355.
• J. Sondow and E. W. Weisstein, MathWorld: Smarandache Function

Crossrefs

Cf. A046021.

Programs

Mathematica

a[n_] := DivisorSigma[0, n!] - DivisorSigma[0, (n-1)!]; a[1] = 1;
Array[a, 36] (* Jean-François Alcover, Sep 17 2020 *)

PARI

a(n)=numdiv(n!)-numdiv((n-1)!) \\ Charles R Greathouse IV, Jul 21 2015

Formula

a(n) = A027423(n)-A027423(n-1) = A000005(A000142(n))-A000005(A000142(n-1)) i.e., number of divisors of n! which are not divisors of (n-1)! [for n>1]. - Henry Bottomley, Oct 22 2001

Erdős, Graham, Ivić, & Pomerance show that the average order of log a(n) is c log n/(log log n)^2 with c around 0.6289. - Charles R Greathouse IV, Jul 21 2015