A382080 Number of ways to partition the prime indices of n into sets with a common sum.

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

Offset

1,30

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, sum A056239.

Also the number of factorizations of n into squarefree numbers > 1 with equal sums of prime indices.

Links

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

Example

The prime indices of 900 are {1,1,2,2,3,3}, with the following partitions into sets with a common sum:
  {{1,2,3},{1,2,3}}
  {{3},{3},{1,2},{1,2}}
So a(900) = 2.

Mathematica

prix[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
sps[{}]:={{}}; sps[set:{i_, ___}]:=Join@@Function[s, Prepend[#, s]& /@ sps[Complement[set, s]]]/@Cases[Subsets[set], {i, ___}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]& /@ sps[Range[Length[set]]]];
Table[Length[Select[mps[prix[n]], SameQ@@Total/@#&&And@@UnsameQ@@@#&]], {n, 100}]

Crossrefs

For just sets we have A050320, distinct A050326.

Twice-partitions of this type are counted by A279788.

For just a common sum we have A321455.

MM-numbers of these multiset partitions are A326534 /\ A302478.

For distinct instead of equal sums we have A381633.

For constant instead of strict blocks we have A381995.

Positions of 0 are A381719, counted by A381994.

A000688 counts factorizations into prime powers, distinct A050361.

A001055 counts factorizations, strict A045778.

A055396 gives least prime index, greatest A061395.

A056239 adds up prime indices, row sums of A112798.

A317141 counts coarsenings of prime indices, refinements A300383.

Cf. A000720, A000961, A001222, A006171, A279784, A353866, A381635, A381871.

Sequence in context: A249351 A123706 A322817 * A194325 A300547 A025452

Adjacent sequences: A382077 A382078 A382079 * A382081 A382082 A382083

Keyword

nonn

Author

Gus Wiseman, Mar 20 2025

Status

approved