A316979 Number of strict factorizations of n into factors > 1 with no equivalent primes.

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

Offset: 1

Author

Gus Wiseman, Jul 18 2018

Keywords

• nonn

Comments

In a factorization, two primes are equivalent if each factor has in its prime factorization the same multiplicity of both primes. For example, in 60 = (2*30) the primes {3, 5} are equivalent but {2, 3} and {2, 5} are not.

Examples

			The a(24) = 5 factorizations are (2*3*4), (2*12), (3*8), (4*6), (24).
The a(36) = 4 factorizations are (2*3*6), (2*18), (3*12), (4*9).
		

Crossrefs

Cf. A000009, A001055, A007716, A007717, A020555, A045778, A130091, A162247, A281116.

Cf. A316974, A316978, A316980, A316981.

Programs

Mathematica

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
dual[eds_]:=Table[First/@Position[eds,x],{x,Union@@eds}];
Table[Length[Select[facs[n],And[UnsameQ@@#,UnsameQ@@dual[primeMS/@#]]&]],{n,100}]

Formula

a(prime^n) = A000009(n).