A386587 Number of ways to choose a pairwise disjoint family of strict integer partitions, one of each exponent in the prime factorization of n.

1, 1, 1, 1, 1, 0, 1, 2, 1, 0, 1, 1, 1, 0, 0, 2, 1, 1, 1, 1, 0, 0, 1, 1, 1, 0, 2, 1, 1, 0, 1, 3, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 4, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 2, 0, 1, 0, 0, 0, 0

Offset: 1

Author

Gus Wiseman, Aug 06 2025

Keywords

• nonn

Comments

First differs from A382525 at a(216) = 1, A382525(216) = 2.

Examples

			The prime exponents of 864 = 2^5 * 3^3 are (5,3), with disjoint families {{3},{5}}, {{3},{1,4}}, {{5},{1,2}}, so a(864) = 3.
		

Crossrefs

Positions of positive terms are A351294, conjugate A381432.

Positions of 0 are A351295, conjugate A381433.

For ordered set partitions we have A382525.

Positions of first appearances are A382775.

The separable case is A386575.

The inseparable case is A386582, see A386632.

A000110 counts set partitions, ordered A000670.

A003242 and A335452 count separations, ranks A333489.

A239455 counts Look-and-Say partitions, complement A351293.

A279790 counts disjoint families on strongly normal multisets.

A325534 counts separable multisets, ranks A335433, sums of A386583.

A325535 counts inseparable multisets, ranks A335448, sums of A386584.

A386633 counts separable set partitions, row sums of A386635.

A386634 counts inseparable set partitions, row sums of A386636.

Cf. A001221, A001222, A008480, A025065, A051903, A051904, A056239, A130091, A336106, A373957, A386580.

Programs

Mathematica

disjointFamilies[y_]:=Union[Sort/@Select[Tuples[IntegerPartitions/@Length/@Split[y]],UnsameQ@@Join@@#&]];
prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[disjointFamilies[prix[n]]],{n,100}]