A384389 Number of proper ways to choose disjoint strict integer partitions of each prime index of n.

0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 2, 0, 3, 0, 0, 0, 4, 0, 5, 0, 1, 1, 7, 0, 2, 1, 0, 0, 9, 0, 11, 0, 1, 2, 1, 0, 14, 2, 1, 0, 17, 0, 21, 0, 0, 4, 26, 0, 2, 0, 2, 0, 31, 0, 2, 0, 3, 4, 37, 0, 45, 6, 0, 0, 3, 0, 53, 0, 4, 0, 63, 0, 75, 7, 0, 0, 2, 0, 88, 0, 0, 9

Offset: 1

Author

Gus Wiseman, Jun 01 2025

Keywords

• nonn

Comments

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

By "proper" we exclude the case of all singletons, which is disjoint when n is squarefree.

Examples

			The prime indices of 65 are {3,6}, and we have proper choices: ((3),(5,1)), ((3),(4,2)), ((2,1),(6)). Hence a(65) = 3.
The prime indices of 175 are {3,3,4}, and we have choices: ((3),(2,1),(4)), ((2,1),(3),(4)), both already proper. Hence a(175) = 2.
		

Crossrefs

Without disjointness we have A357982 - 1, non-strict version A299200 - 1.

This is the proper case of A383706, conjugate version A384005.

Positions of positive terms are A384321.

Positions of 0 are A384349.

Positions of 1 are A384390.

Positions of terms > 1 are A384393.

The conjugate version is A384394.

Positions of first appearances are A384396.

A000041 counts integer partitions, strict A000009.

A048767 is the Look-and-Say transform, fixed points A048768, counted by A217605.

A055396 gives least prime index, greatest A061395.

A056239 adds up prime indices, row sums of A112798.

A239455 counts Look-and-Say partitions, ranks A351294

A351293 counts non-Look-and-Say partitions, ranks A351295.

Cf. A382912, A383710, A383711, A382913, A383708, A383533.

Cf. A179009, A279375, A279790, A381454, A382525, A384322, A384347.

Programs

Mathematica

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

Formula

a(prime(n)) = A000009(n) - 1.