A384005 Number of ways to choose disjoint strict integer partitions, one of each conjugate prime index of n.

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

Offset: 1

Author

Gus Wiseman, May 22 2025

Keywords

• nonn

Examples

			The prime indices of 96 are {1,1,1,1,1,2}, conjugate (6,1), and we have choices (6,1) and (4,2,1), so a(96) = 2.
The prime indices of 108 are {1,1,2,2,2}, conjugate (5,3), and we have choices (5,3), (5,2,1), (4,3,1), so a(108) = 3.
		

Crossrefs

Adding up over all integer partitions gives A279790, strict A279375.

For multiplicities instead of indices we have conjugate of A382525.

The conjugate version is A383706.

Positive positions are A384010, conjugate A382913, counted by A383708, odd case A383533.

Positions of 0 are A384011.

Without disjointness we have A384179, conjugate A357982, non-strict version A299200.

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 or section-sum partitions, ranks A351294 or A381432.

A351293 counts non Look-and-Say or non section-sum partitions, ranks A351295 or A381433.

Cf. `A098859, A122111, A130091, A179009, A317141, A382771, A383707, A383710.

Programs

Mathematica

pof[y_]:=Select[Join@@@Tuples[IntegerPartitions/@y],UnsameQ@@#&];
conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[pof[conj[prix[n]]]],{n,100}]

Formula

a(n) = A383706(A122111(n)).