A336424 Number of factorizations of n where each factor belongs to A130091 (numbers with distinct prime multiplicities).

1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 3, 1, 1, 1, 5, 1, 3, 1, 3, 1, 1, 1, 5, 2, 1, 3, 3, 1, 1, 1, 7, 1, 1, 1, 6, 1, 1, 1, 5, 1, 1, 1, 3, 3, 1, 1, 9, 2, 3, 1, 3, 1, 5, 1, 5, 1, 1, 1, 4, 1, 1, 3, 11, 1, 1, 1, 3, 1, 1, 1, 11, 1, 1, 3, 3, 1, 1, 1, 9, 5, 1, 1, 4, 1, 1

Offset

1,4

Comments

A number's prime signature (row n of A124010) is the sequence of positive exponents in its prime factorization, so a number has distinct prime multiplicities iff all the exponents in its prime signature are distinct.

Links

Table of n, a(n) for n=1..86.

Example

The a(n) factorizations for n = 2, 4, 8, 60, 16, 36, 32, 48:
  2  4    8      5*12     16       4*9      32         48
     2*2  2*4    3*20     4*4      3*12     4*8        4*12
          2*2*2  3*4*5    2*8      3*3*4    2*16       3*16
                 2*2*3*5  2*2*4    2*18     2*4*4      3*4*4
                          2*2*2*2  2*2*9    2*2*8      2*24
                                   2*2*3*3  2*2*2*4    2*3*8
                                            2*2*2*2*2  2*2*12
                                                       2*2*3*4
                                                       2*2*2*2*3

Mathematica

facsusing[s_, n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[facsusing[Select[s, Divisible[n/d, #]&], n/d], Min@@#>=d&]], {d, Select[s, Divisible[n, #]&]}]];
Table[Length[facsusing[Select[Range[2, n], UnsameQ@@Last/@FactorInteger[#]&], n]], {n, 100}]

Crossrefs

A327523 is the case when n is restricted to belong to A130091 also.

A001055 counts factorizations.

A007425 counts divisors of divisors.

A045778 counts strict factorizations.

A074206 counts ordered factorizations.

A130091 lists numbers with distinct prime multiplicities.

A181796 counts divisors with distinct prime multiplicities.

A253249 counts nonempty chains of divisors.

A281116 counts factorizations with no common divisor.

A302696 lists numbers whose prime indices are pairwise coprime.

A305149 counts stable factorizations.

A320439 counts factorizations using A289509.

A327498 gives the maximum divisor with distinct prime multiplicities.

A336500 counts divisors of n in A130091 with quotient also in A130091.

A336568 = not a product of two numbers with distinct prime multiplicities.

A336569 counts maximal chains of elements of A130091.

A337256 counts chains of divisors.

Cf. A071625, A080688, A098859, A118914, A124010, A167865, A294068, A303707, A305150, A322453, A336420, A336570, A336571.

Sequence in context: A136565 A181591 A347442 * A353236 A325939 A318586

Adjacent sequences: A336421 A336422 A336423 * A336425 A336426 A336427

Keyword

nonn

Author

Gus Wiseman, Aug 03 2020

Status

approved