A383706 - OEIS

A383706

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

1, 1, 1, 0, 2, 1, 2, 0, 0, 1, 3, 0, 4, 1, 1, 0, 5, 0, 6, 0, 2, 2, 8, 0, 2, 2, 0, 0, 10, 1, 12, 0, 2, 3, 2, 0, 15, 3, 2, 0, 18, 1, 22, 0, 0, 5, 27, 0, 2, 0, 3, 0, 32, 0, 3, 0, 4, 5, 38, 0, 46, 7, 0, 0, 4, 1, 54, 0, 5, 1, 64, 0, 76, 8, 0, 0, 3, 1, 89, 0, 0, 10

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OFFSET

1,5

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..82.

EXAMPLE

The prime indices of 25 are (3,3), for which we have choices ((3),(2,1)) and ((2,1),(3)), so a(25) = 2.

The prime indices of 91 are (4,6), for which we have choices ((4),(6)), ((4),(5,1)), ((4),(3,2,1)), ((3,1),(6)), ((3,1),(4,2)), so a(91) = 5.

The prime indices of 273 are (2,4,6), for which we have choices ((2),(4),(6)), ((2),(4),(5,1)), ((2),(3,1),(6)), so a(273) = 3.

MATHEMATICA

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

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

Table[Length[pof[prix[n]]], {n, 100}]

CROSSREFS

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

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

For multiplicities instead of indices we have A382525.

Positions of 0 appear to be A382912, counted by A383710, odd case A383711.

Positions of positive terms are A382913, counted by A383708, odd case A383533.

Positions of 1 are A383707, counted by A179009.

The conjugate version 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. A098859, A122111, A130091, A209229, A317141, A381454, A382771, A382876.

Sequence in context: A348920 A348921 A050601 * A101650 A287411 A053796

Adjacent sequences: A383703 A383704 A383705 * A383707 A383708 A383709

KEYWORD

nonn

AUTHOR

Gus Wiseman, May 15 2025

STATUS

approved