A321469 Number of factorizations of n into factors > 1 with different sums of prime indices. Number of multiset partitions of the multiset of prime indices of n with distinct block-sums.

1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 2, 2, 1, 3, 1, 3, 2, 2, 1, 4, 1, 2, 2, 3, 1, 4, 1, 3, 2, 2, 2, 5, 1, 2, 2, 4, 1, 5, 1, 3, 3, 2, 1, 6, 1, 3, 2, 3, 1, 5, 2, 5, 2, 2, 1, 7, 1, 2, 2, 4, 2, 5, 1, 3, 2, 4, 1, 8, 1, 2, 3, 3, 2, 5, 1, 6, 2, 2, 1, 7, 2, 2, 2

Offset

1,6

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. The sum of prime indices of n is A056239(n).

Links

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

Gus Wiseman, Sequences counting and ranking multiset partitions whose part lengths, sums, or averages are constant or strict.

Example

The a(72) = 8 multiset partitions with distinct block-sums:
    {{1,1,1,2,2}}
   {{1},{1,1,2,2}}
   {{2},{1,1,1,2}}
   {{1,1},{1,2,2}}
   {{1,2},{1,1,2}}
   {{2,2},{1,1,1}}
  {{1},{2},{1,1,2}}
  {{1},{1,1},{2,2}}
Missing from this list are:
    {{1},{1},{1,2,2}}
    {{1},{1,2},{1,2}}
    {{2},{2},{1,1,1}}
    {{2},{1,1},{1,2}}
   {{1},{1},{1},{2,2}}
   {{1},{1},{2},{1,2}}
   {{1},{2},{2},{1,1}}
  {{1},{1},{1},{2},{2}}

Mathematica

primeMS[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[primeMS[n]], UnsameQ@@Sort[Total/@#]&]], {n, 100}]

Crossrefs

Cf. A001055, A275780, A317144, A321455, A322794, A326514, A326515, A326516, A326519, A326535.

Sequence in context: A334461 A338652 A033109 * A365460 A235644 A298475

Adjacent sequences: A321466 A321467 A321468 * A321470 A321471 A321472

Keyword

nonn

Author

Gus Wiseman, Nov 11 2018

Status

approved