A336425 Number of ways to choose a divisor with distinct prime exponents of a divisor with distinct prime exponents of n!.

1, 1, 3, 5, 24, 38, 132, 195, 570, 1588, 4193, 6086, 14561, 19232, 37142, 106479, 207291, 266871, 549726, 674330, 1465399, 3086598, 5939574, 7182133, 12324512, 28968994, 46819193, 82873443, 165205159, 196666406, 350397910, 406894074, 593725529, 1229814478, 1853300600, 4024414209, 6049714096, 6968090487, 9700557121, 16810076542, 26339337285

Offset: 0

Author

Gus Wiseman, Aug 06 2020

Keywords

• nonn

Examples

			The a(4) = 24 divisors of divisors:
  1/1  2/1  3/1  4/1  8/1  12/1   24/1
       2/2  3/3  4/2  8/2  12/2   24/2
                 4/4  8/4  12/3   24/3
                      8/8  12/4   24/4
                           12/12  24/8
                                  24/12
                                  24/24
		

Links

• Max Alekseyev, Table of n, a(n) for n = 0..85

Crossrefs

A336422 is the non-factorial generalization.

A130091 lists numbers with distinct prime exponents.

A181796 counts divisors with distinct prime exponents.

A327526 gives the maximum divisor of n with equal prime exponents.

A327498 gives the maximum divisor of n with distinct prime exponents.

A336414 counts divisors of n! with distinct prime exponents.

A336415 counts divisors of n! with equal prime exponents.

A336423 counts chains in A130091, with maximal version A336569.

Cf. A000005, A000110, A098859, A118914, A124010, A336424, A336500, A336568, A336570, A336571, A336865, A336866, A336869.

Factorial numbers: A000142, A022559, A027423, A048656, A048742, A071626, A325272, A325273, A325617, A327499, A336416, A336418, A336617.

Programs

Mathematica

strsigQ[n_]:=UnsameQ@@Last/@FactorInteger[n];
Table[Total[Cases[Divisors[n!],d_?strsigQ:>Count[Divisors[d],e_?strsigQ]]],{n,0,20}]

Extensions

Terms a(21) onward from Max Alekseyev, Nov 07 2024