A336414 Number of divisors of n! with distinct prime multiplicities.

1, 1, 2, 3, 7, 10, 20, 27, 48, 86, 147, 195, 311, 390, 595, 1031, 1459, 1791, 2637, 3134, 4747, 7312, 10766, 12633, 16785, 26377, 36142, 48931, 71144, 82591, 112308, 128023, 155523, 231049, 304326, 459203, 568095, 642446, 812245, 1137063, 1441067, 1612998, 2193307, 2429362

Offset: 0

Author

Gus Wiseman, Jul 22 2020

Keywords

• nonn

Comments

A number has distinct prime multiplicities iff its prime signature is strict.

Examples

			The first and second columns below are the a(6) = 20 counted divisors of 6! together with their prime signatures. The third column shows the A000005(6!) - a(6) = 10 remaining divisors.
      1: ()      20: (2,1)    |    6: (1,1)
      2: (1)     24: (3,1)    |   10: (1,1)
      3: (1)     40: (3,1)    |   15: (1,1)
      4: (2)     45: (2,1)    |   30: (1,1,1)
      5: (1)     48: (4,1)    |   36: (2,2)
      8: (3)     72: (3,2)    |   60: (2,1,1)
      9: (2)     80: (4,1)    |   90: (1,2,1)
     12: (2,1)  144: (4,2)    |  120: (3,1,1)
     16: (4)    360: (3,2,1)  |  180: (2,2,1)
     18: (1,2)  720: (4,2,1)  |  240: (4,1,1)
		

Links

• Chai Wah Wu, Table of n, a(n) for n = 0..6245 (n = 0..94 from David A. Corneth)

Crossrefs

Perfect-powers are A001597, with complement A007916.

Numbers with distinct prime multiplicities are A130091.

Divisors with distinct prime multiplicities are counted by A181796.

The maximum divisor with distinct prime multiplicities is A327498.

Divisors of n! with equal prime multiplicities are counted by A336415.

Cf. A000005, A098859, A118914, A124010, A327527, A336424, A336500, A336568.

Factorial numbers: A000142, A007489, A022559, A027423, A048656, A048742, A071626, A325272, A325273, A325617, A336416.

Programs

Mathematica

Table[Length[Select[Divisors[n!],UnsameQ@@Last/@FactorInteger[#]&]],{n,0,15}]

PARI

a(n) = sumdiv(n!, d, my(ex=factor(d)[,2]); #vecsort(ex,,8) == #ex); \\ Michel Marcus, Jul 24 2020

Formula

a(n) = A181796(n!).

Extensions

a(21)-a(41) from Alois P. Heinz, Jul 24 2020