A316979 - OEIS

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

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OFFSET

1,8

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.

LINKS

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

FORMULA

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

EXAMPLE

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).

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}]