A356945 Number of multiset partitions of the prime indices of n such that each block covers an initial interval. Number of factorizations of n into members of A055932.

1, 1, 0, 2, 0, 1, 0, 3, 0, 0, 0, 2, 0, 0, 0, 5, 0, 1, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 1, 0, 7, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 7, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 2, 0, 0, 0, 11, 0, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Author

Gus Wiseman, Sep 08 2022

Keywords

• nonn

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.

Examples

			The a{n} multiset partitions for n = 8, 24, 72, 96:
  {{111}}      {{1112}}      {{11122}}      {{111112}}
  {{1}{11}}    {{1}{112}}    {{1}{1122}}    {{1}{11112}}
  {{1}{1}{1}}  {{11}{12}}    {{11}{122}}    {{11}{1112}}
               {{1}{1}{12}}  {{12}{112}}    {{111}{112}}
                             {{1}{1}{122}}  {{12}{1111}}
                             {{1}{12}{12}}  {{1}{1}{1112}}
                                            {{1}{11}{112}}
                                            {{11}{11}{12}}
                                            {{1}{12}{111}}
                                            {{1}{1}{1}{112}}
                                            {{1}{1}{11}{12}}
                                            {{1}{1}{1}{1}{12}}
		

Links

• Gus Wiseman, Counting and ranking classes of multiset partitions related to gapless multisets

Crossrefs

Positions of 0's are A080259, complement A055932.

A000688 counts factorizations into prime powers.

A001055 counts factorizations.

A001221 counts prime divisors, with sum A001414.

A001222 counts prime factors with multiplicity.

A056239 adds up prime indices, row sums of A112798.

A356069 counts gapless divisors, initial A356224 (complement A356225).

Multisets covering an initial interval are counted by A000009, A000041, A011782, ranked by A055932.

Other types: A034691, A089259, A356954, A356955.

Other conditions: A050320, A050330, A322585, A356233, A356931, A356936.

Cf. A003963, A073491, A107742, A287170, A324929, A340852, A356234.

Programs

Mathematica

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
nnQ[m_]:=PrimePi/@First/@FactorInteger[m]==Range[PrimePi[Max@@First/@FactorInteger[m]]];
Table[Length[Select[facs[n],And@@nnQ/@#&]],{n,100}]