A327400 Number of factorizations of n that are constant or whose factors are relatively prime.

1, 1, 1, 2, 1, 2, 1, 2, 2, 2, 1, 3, 1, 2, 2, 3, 1, 3, 1, 3, 2, 2, 1, 4, 2, 2, 2, 3, 1, 5, 1, 2, 2, 2, 2, 7, 1, 2, 2, 4, 1, 5, 1, 3, 3, 2, 1, 6, 2, 3, 2, 3, 1, 4, 2, 4, 2, 2, 1, 9, 1, 2, 3, 4, 2, 5, 1, 3, 2, 5, 1, 10, 1, 2, 3, 3, 2, 5, 1, 6, 3, 2, 1, 9, 2, 2, 2

Offset

1,4

Comments

First differs from A327399 at a(24) = 4, A327399(24) = 3.

Links

Table of n, a(n) for n=1..87.

Formula

a(n) = A281116(n) + A089723(n).

Example

The factorizations of 2, 4, 12, 24, 30, 36, 48, and 60 that are constant or whose factors are relatively prime:
  2   4     12      24        30      36        48          60
      2*2   3*4     3*8       5*6     4*9       3*16        3*20
            2*2*3   2*3*4     2*15    6*6       2*3*8       4*15
                    2*2*2*3   3*10    2*2*9     3*4*4       5*12
                              2*3*5   2*3*6     2*2*3*4     2*5*6
                                      3*3*4     2*2*2*2*3   3*4*5
                                      2*2*3*3               2*2*15
                                                            2*3*10
                                                            2*2*3*5

Mathematica

facs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[facs[n/d], Min@@#>=d&]], {d, Rest[Divisors[n]]}]];
Table[Length[Select[facs[n], #=={}||Length[Union[#]]==1||GCD@@#==1&]], {n, 100}]

Crossrefs

Constant factorizations are A089723.

Cf. A007359, A051424, A281116, A302569, A304709, A304711, A319269, A327399.

Sequence in context: A320732 A305149 A336737 * A323303 A068108 A342679

Adjacent sequences: A327397 A327398 A327399 * A327401 A327402 A327403

Keyword

nonn

Author

Gus Wiseman, Sep 22 2019

Status

approved