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

0,4

Comments

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

Links

Table of n, a(n) for n=0..38.

Formula

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

a(n) = A070824(n!).

Example

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

Mathematica

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

Prog

(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

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.

Sequence in context: A290699 A027083 A249665 * A321027 A214907 A169948

Adjacent sequences: A337103 A337104 A337105 * A337107 A337108 A337109

Keyword

nonn

Author

Gus Wiseman, Aug 23 2020

Extensions

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

Status

approved