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

1,4

Comments

In a factorization, two primes are equivalent if each factor has in its prime factorization the same multiplicity of both primes.

Links

Table of n, a(n) for n=1..86.

Formula

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

a(squarefree) = 1.

Example

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).

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}]

Crossrefs

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

Cf. A316974, A316979, A316980, A316981.

Sequence in context: A318586 A222580 A376140 * A331023 A284345 A347463

Adjacent sequences: A316975 A316976 A316977 * A316979 A316980 A316981

Keyword

nonn

Author

Gus Wiseman, Jul 18 2018

Status

approved