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

0,3

Comments

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

Links

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

Formula

a(n) = A181796(n!).

Example

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)

Mathematica

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

Prog

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

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.

Sequence in context: A173132 A351704 A320675 * A095010 A295723 A306008

Adjacent sequences: A336411 A336412 A336413 * A336415 A336416 A336417

Keyword

nonn

Author

Gus Wiseman, Jul 22 2020

Extensions

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

Status

approved