A027423 Number of divisors of n!.

1, 1, 2, 4, 8, 16, 30, 60, 96, 160, 270, 540, 792, 1584, 2592, 4032, 5376, 10752, 14688, 29376, 41040, 60800, 96000, 192000, 242880, 340032, 532224, 677376, 917280, 1834560, 2332800, 4665600, 5529600, 7864320, 12165120, 16422912

Offset: 0

Author

Glen Burch (gburch(AT)erols.com), Leroy Quet

Keywords

• nonn
• easy
• nice

Comments

It appears that a(n+1)=2*a(n) if n is in A068499. - Benoit Cloitre, Sep 07 2002

Because a(0) = 1 and for all n > 0, 2*a(n) >= a(n+1), the sequence is a complete sequence. - Frank M Jackson, Aug 09 2013

Luca and Young prove that a(n) divides n! for n >= 6. - Michel Marcus, Nov 02 2017

Examples

			a(4) = 8 because 4!=24 has precisely eight distinct divisors: 1, 2, 3, 4, 6, 8, 12, 24.
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from T. D. Noe)
• Daniel Berend and J. E. Harmse, Gaps between consecutive divisors of factorials, Ann. Inst. Fourier, 43 (3) (1993), 569-583.
• 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.
• Florian Luca and Paul Thomas Young, On the number of divisors of n! and of the Fibonacci numbers, Glasnik Matematicki, Vol. 47, No. 2 (2012), 285-293. DOI: 10.3336/gm.47.2.05.
• John D. Mahony, The next number in the sequence, Math. Gaz. (2024) Vol. 108, Issue 573, 399-406.
• Wikipedia, Complete sequence.
• Index entries for sequences related to factorial numbers
• Index entries for sequences related to divisors of numbers

Crossrefs

Cf. A000005, A000142, A062569, A131688, A161466 (divisors of 10!).

Programs

Haskell

a027423 n = f 1 $ map (\p -> iterate (* p) p) a000040_list where
   f y ((pps@(p:_)):ppss)
     | p <= n = f (y * (sum (map (div n) $ takeWhile (<= n) pps) + 1)) ppss
     | otherwise = y
-- Reinhard Zumkeller, Feb 27 2013
(Python 3.8+)
from math import prod
from collections import Counter
from sympy import factorint
def A027423(n): return prod(e+1 for e in sum((Counter(factorint(i)) for i in range(2,n+1)),start=Counter()).values()) # Chai Wah Wu, Jun 25 2022

Maple

A027423 := n -> numtheory[tau](n!);

Mathematica

Table[ DivisorSigma[0, n! ], {n, 0, 35}]

PARI

for(k=0,50,print1(numdiv(k!),", ")) \\ Jaume Oliver Lafont, Mar 09 2009

PARI

a(n)=my(s=1,t,tt);forprime(p=2,n,t=tt=n\p; while(tt, t+=tt\=p); s*=t+1); s \\ Charles R Greathouse IV, Feb 08 2013

Formula

a(n) <= a(n+1) <= 2*a(n) - Benoit Cloitre, Sep 07 2002

From Avik Roy (avik_3.1416(AT)yahoo.co.in), Jan 28 2009: (Start)

Assume, p1,p2...pm are the prime numbers less than or equal to n.

Then, a(n) = Product_{i=1..m} (bi+1), where bk = Sum_{i=1..m} floor(n/pk^i).

For example, if n=5, p1=2,p2=3,p3=5;

b1=floor(5/2)+floor(5/2^2)+floor(5/2^3)+...=2+1+0+..=3 similarly, b2=b3=1;

Thus a(5)=(3+1)(1+1)(1+1)=16. (End)

a(n) = A000005(A000142(n)). - Michel Marcus, Sep 13 2014

a(n) ~ exp(c * n/log(n) + O(n/log(n)^2)), where c = A131688 (Erdős et al., 1996). - Amiram Eldar, Nov 07 2020