A354911 Number of factorizations of n into relatively prime prime-powers.

1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 2, 0, 1, 1, 0, 0, 2, 0, 2, 1, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 0, 1, 1, 1, 4, 0, 1, 1, 3, 0, 1, 0, 2, 2, 1, 0, 5, 0, 2, 1, 2, 0, 3, 1, 3, 1, 1, 0, 2, 0, 1, 2, 0, 1, 1, 0, 2, 1, 1, 0, 6, 0, 1, 2, 2, 1, 1, 0, 5, 0, 1, 0, 2, 1, 1, 1

Offset: 1

Author

Gus Wiseman, Jul 25 2022

Keywords

• nonn

Examples

			The a(n) factorizations for n = 6, 12, 24, 36, 48, 72, 96:
  2*3  3*4    3*8      4*9      3*16       8*9        3*32
       2*2*3  2*3*4    2*2*9    2*3*8      2*4*9      3*4*8
              2*2*2*3  3*3*4    3*4*4      3*3*8      2*3*16
                       2*2*3*3  2*2*3*4    2*2*2*9    2*2*3*8
                                2*2*2*2*3  2*3*3*4    2*3*4*4
                                           2*2*2*3*3  2*2*2*3*4
                                                      2*2*2*2*2*3
		

Links

• Wikipedia, Coprime integers.

Crossrefs

This is the relatively prime case of A000688, partitions A023894.

Positions of 0's are A246655 (A000961 includes 1).

For strict instead of relatively prime we have A050361, partitions A054685.

Positions of 1's are A000469 (A120944 excludes 1).

For pairwise coprime instead of relatively prime we have A143731.

The version for partitions instead of factorizations is A356067.

A000005 counts divisors.

A001055 counts factorizations.

A001221 counts distinct prime divisors, with sum A001414.

A001222 counts prime-power divisors.

A289509 lists numbers whose prime indices are relatively prime.

A295935 counts twice-factorizations with constant blocks (type PPR).

A355743 lists numbers with prime-power prime indices, squarefree A356065.

Cf. A000837, A023893, A076610, A085970, A106244, A279784, A318721, A355737, A355742, A356064, A356066.

Programs

Mathematica

ufacs[s_,n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&, Select[ufacs[Select[s,Divisible[n/d,#]&],n/d],Min@@#>=d&]],{d,Select[s,Divisible[n,#]&]}]];
Table[Length[Select[ufacs[Select[Divisors[n],PrimePowerQ[#]&],n],GCD@@#<=1&]],{n,100}]

Formula

a(n) = A000688(n) if n is nonprime, otherwise a(n) = 0.