A384179 Number of ways to choose strict integer partitions of each conjugate prime index of n.

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

Offset

1,8

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.

Links

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

Example

The prime indices of 180 are {1,1,2,2,3}, conjugate {5,3,1}, and we have choices:
  {{5},{3},{1}}
  {{5},{2,1},{1}}
  {{4,1},{3},{1}}
  {{4,1},{2,1},{1}}
  {{3,2},{3},{1}}
  {{3,2},{2,1},{1}}
so a(180) = 6.

Mathematica

fop[y_]:=Join@@@Tuples[strptns/@y];
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[fop[conj[prix[n]]]], {n, 100}]

Crossrefs

Positions of 1 are A037143, complement A033942.

For multiplicities instead of indices we have A050361.

Adding up over all integer partitions gives A270995, disjoint A279790, strict A279375.

The conjugate version is A357982, disjoint A383706.

The disjoint case is A384005.

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. A122111, A130091, A179009, A382913, A383707, A383710, A384010, A384011.

Sequence in context: A347441 A360617 A102097 * A387537 A361566 A354991

Adjacent sequences: A384176 A384177 A384178 * A384180 A384181 A384182

Keyword

nonn

Author

Gus Wiseman, May 23 2025

Status

approved