A384394 - OEIS

A384394

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

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

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OFFSET

1,27

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.

LINKS

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

EXAMPLE

The prime indices of 216 are {1,1,1,2,2,2}, with conjugate partition (6,3), with proper choices ((6),(2,1)), ((5,1),(3)), and ((4,2),(3)), so a(216) = 3.

MATHEMATICA

prix[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];

conj[y_]:=If[Length[y]==0, y, Table[Length[Select[y, #>=k&]], {k, 1, Max[y]}]];

pofprop[y_]:=Select[DeleteCases[Join@@@Tuples[IntegerPartitions/@y], y], UnsameQ@@#&];

Table[Length[pofprop[conj[prix[n]]]], {n, 100}]

CROSSREFS

Conjugate prime indices are the rows of A122111.

The non-proper version is A384005, conjugate A383706.

This is the conjugate version of A384389 (firsts 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 or section-sum partitions, ranks A351294 or A381432.

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

See also A382912, counted by A383710, odd case A383711.

See also A382913, counted by A383708, odd case A383533.

Cf. A279790, A357982, A381454, A382525, A384321, A384322, A384347, A384349, A384390, A384393.

Sequence in context: A394206 A234859 A001343 * A394203 A022882 A171919

Adjacent sequences: A384391 A384392 A384393 * A384395 A384396 A384397

KEYWORD

nonn

AUTHOR

Gus Wiseman, Jun 03 2025

STATUS

approved