A153823 Number of proper divisors of n!.

0, 0, 1, 3, 7, 15, 29, 59, 95, 159, 269, 539, 791, 1583, 2591, 4031, 5375, 10751, 14687, 29375, 41039, 60799, 95999, 191999, 242879, 340031, 532223, 677375, 917279, 1834559, 2332799, 4665599, 5529599, 7864319, 12165119, 16422911

Offset: 0

Author

Omar E. Pol, Jan 02 2009

Keywords

• nonn

Comments

a(n) is the number of proper divisors of factorial number A000142(n).

Examples

			For n=4, 4! = 4*3*2*1 = 24, which has 7 proper divisors: 1, 2, 3, 4, 6, 8, and 12.  So a(4) = 7. - _Michael B. Porter_, Aug 30 2016
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A000005, A000142, A027423, A032741.

Cf. A153823, A153825. [From Omar E. Pol, Jan 17 2009]

Programs

Magma

[DivisorSigma(0,Factorial(n)) - 1: n in [0..40]]; // Vincenzo Librandi, Aug 31 2016

Mathematica

Table[DivisorSigma[0, n!] - 1, {n, 0, 50}] (* G. C. Greubel, Aug 30 2016 *)

PARI

a(n) = numdiv(n!) - 1; \\ Michel Marcus, Aug 31 2016

Python

from sympy import factorial, divisor_count
def A153823(n):
    return divisor_count(factorial(n))-1 # Chai Wah Wu, Aug 24 2020

Formula

a(n) = A000005(A000142(n)) - 1 = A032741(A000142(n)) = A027423(n) - 1.

Extensions

More terms from Omar E. Pol, Jan 17 2009