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

1,3

Links

Table of n, a(n) for n=1..36.

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

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

Mathematica

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

Prog

(PARI) a(n)=numdiv(n!)-numdiv((n-1)!) \\ Charles R Greathouse IV, Jul 21 2015
(Python)
from math import prod
from collections import Counter
from sympy import factorint
def A038024(n):
    if n == 1: return 1
    a = sum((Counter(factorint(i)) for i in range(2, n)), start=Counter())
    return prod(e+1 for e in (a+Counter(factorint(n))).values())-prod(e+1 for e in a.values()) # Chai Wah Wu, Feb 23 2026

Crossrefs

Cf. A046021.

Sequence in context: A244933 A118560 A187813 * A337500 A388287 A061297

Adjacent sequences: A038021 A038022 A038023 * A038025 A038026 A038027

Keyword

nonn

Author

Christian G. Bower

Status

approved