A316978 Number of factorizations of n into factors > 1 with no equivalent primes.

1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 4, 1, 1, 1, 5, 1, 4, 1, 4, 1, 1, 1, 7, 2, 1, 3, 4, 1, 1, 1, 7, 1, 1, 1, 7, 1, 1, 1, 7, 1, 1, 1, 4, 4, 1, 1, 12, 2, 4, 1, 4, 1, 7, 1, 7, 1, 1, 1, 7, 1, 1, 4, 11, 1, 1, 1, 4, 1, 1, 1, 16, 1, 1, 4, 4, 1, 1, 1, 12, 5, 1, 1, 7, 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.

Examples

			The a(36) = 7 factorizations are (2*2*3*3), (2*2*9), (2*3*6), (3*3*4), (2*18), (3*12), (4*9). Missing from this list are (6*6) and (36).
		

Crossrefs

Cf. A001055, A007716, A007717, A020555, A045778, A162247, A281116, A303386.

Cf. A316974, A316979, 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],UnsameQ@@dual[primeMS/@#]&]],{n,100}]

Formula

a(prime^n) = A000041(n).

a(squarefree) = 1.