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

Author

Gus Wiseman, Sep 22 2019

Keywords

• nonn

Comments

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

Examples

			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
		

Crossrefs

Constant factorizations are A089723.

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

Programs

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}]

Formula

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