A337106 Number of nontrivial divisors of n!.

0, 0, 0, 2, 6, 14, 28, 58, 94, 158, 268, 538, 790, 1582, 2590, 4030, 5374, 10750, 14686, 29374, 41038, 60798, 95998, 191998, 242878, 340030, 532222, 677374, 917278, 1834558, 2332798, 4665598, 5529598, 7864318, 12165118, 16422910, 19595518, 39191038, 60466174

Offset: 0

Author

Gus Wiseman, Aug 23 2020

Keywords

• nonn

Comments

A divisor of n is trivial if it is 1 or n.

Examples

			The a(3) = 2 through a(5) =14 nontrivial divisions:
  6/2  24/2   120/2
  6/3  24/3   120/3
       24/4   120/4
       24/6   120/5
       24/8   120/6
       24/12  120/8
              120/10
              120/12
              120/15
              120/20
              120/24
              120/30
              120/40
              120/60
		

Crossrefs

A070824 counts nontrivial divisors.

A153823 counts proper divisors of n!.

A337107 has this sequence as column k = 3.

A000005 counts divisors.

A000142 lists factorial numbers.

A001055 counts factorizations.

A027423 counts divisors of factorial numbers.

A067824 counts chains of divisors starting with n.

A074206 counts chains of divisors from n to 1.

A076716 counts factorizations of factorial numbers.

A253249 counts chains of divisors.

A337071 counts chains of divisors starting with n!.

A337105 counts chains of divisors from n! to 1.

Cf. A022559, A124010, A251683, A325617, A336941.

Programs

Mathematica

Table[Length[DeleteCases[Divisors[n!],1|n!]],{n,10}]

Python

from sympy import factorial, divisor_count
def A337106(n):
    return 0 if n <= 1 else divisor_count(factorial(n))-2 # Chai Wah Wu, Aug 24 2020

Formula

a(n) = A000005(n!) - 2 for n > 1.

a(n) = A070824(n!).

Extensions

a(0) from Chai Wah Wu, Aug 24 2020