A322454 Number of multiset partitions with no constant parts of a multiset whose multiplicities are the prime indices of n.

1, 0, 0, 1, 0, 1, 0, 1, 2, 1, 0, 2, 0, 1, 2, 4, 0, 4, 0, 3, 3, 1, 0, 7, 4, 1, 9, 4, 0, 7, 0, 11, 3, 1, 5, 15, 0, 1, 4, 11

Offset

1,9

Comments

This multiset (row n of A305936) is generally not the same as the multiset of prime indices of n. For example, the prime indices of 12 are {1,1,2}, while a multiset whose multiplicities are {1,1,2} is {1,1,2,3}.

Links

Table of n, a(n) for n=1..40.

Example

The a(30) = 7 multiset partitions:
    {{1,1,1,2,2,3}}
   {{1,2},{1,1,2,3}}
   {{1,3},{1,1,2,2}}
   {{2,3},{1,1,1,2}}
   {{1,1,2},{1,2,3}}
   {{1,1,3},{1,2,2}}
  {{1,2},{1,2},{1,3}}

Mathematica

nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]], {#1}]&, If[n==1, {}, Flatten[Cases[FactorInteger[n]//Reverse, {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[nrmptn[n]], Min@@Length/@Union/@#>1&]], {n, 20}]

Crossrefs

Cf. A000688, A000961, A001055, A001597, A023893, A023894, A181821, A318284, A320322, A321407, A321760, A322260, A322452.

Sequence in context: A036577 A317189 A220136 * A133693 A002325 A129134

Adjacent sequences: A322451 A322452 A322453 * A322455 A322456 A322457

Keyword

nonn,more

Author

Gus Wiseman, Dec 09 2018

Status

approved