cp's OEIS Frontend

A383310 Number of ways to choose a strict multiset partition of a factorization of n into factors > 1.

%I A383310 #10 Apr 26 2025 15:27:20
%S A383310 1,1,1,2,1,3,1,5,2,3,1,8,1,3,3,9,1,8,1,8,3,3,1,20,2,3,5,8,1,12,1,19,3,
%T A383310 3,3,24,1,3,3,20,1,12,1,8,8,3,1,46,2,8,3,8,1,20,3,20,3,3,1,38,1,3,8,
%U A383310 37,3,12,1,8,3,12,1,67,1,3,8,8,3,12,1,46,9,3
%N A383310 Number of ways to choose a strict multiset partition of a factorization of n into factors > 1.
%e A383310 The a(36) = 24 choices:
%e A383310   {{2,2,3,3}}  {{2},{2,3,3}}  {{2},{3},{2,3}}
%e A383310   {{2,2,9}}    {{3},{2,2,3}}  {{2},{3},{6}}
%e A383310   {{2,3,6}}    {{2,2},{3,3}}
%e A383310   {{2,18}}     {{2},{2,9}}
%e A383310   {{3,3,4}}    {{9},{2,2}}
%e A383310   {{3,12}}     {{2},{3,6}}
%e A383310   {{4,9}}      {{3},{2,6}}
%e A383310   {{6,6}}      {{6},{2,3}}
%e A383310   {{36}}       {{2},{18}}
%e A383310                {{3},{3,4}}
%e A383310                {{4},{3,3}}
%e A383310                {{3},{12}}
%e A383310                {{4},{9}}
%t A383310 facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
%t A383310 sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t A383310 mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
%t A383310 Table[Sum[Length[Select[mps[y],UnsameQ@@#&]],{y,facs[n]}],{n,30}]
%Y A383310 The case of a unique choice (positions of 1) is A008578.
%Y A383310 This is the strict case of A050336.
%Y A383310 For distinct strict blocks we have A050345.
%Y A383310 For integer partitions we have A261049, strict case of A001970.
%Y A383310 For strict blocks that are not necessarily distinct we have A296119.
%Y A383310 Twice-partitions of this type are counted by A296122.
%Y A383310 For normal multisets we have A317776, strict case of A255906.
%Y A383310 A001055 counts factorizations, strict A045778.
%Y A383310 A050320 counts factorizations into squarefree numbers, distinct A050326.
%Y A383310 A281113 counts twice-factorizations, strict A296121, see A296118, A296120.
%Y A383310 Cf. A000009, A005117, A045782, A050342, A279785, A293243, A293511, A302494, A316439, A358914, A381992, A382201.
%K A383310 nonn
%O A383310 1,4
%A A383310 _Gus Wiseman_, Apr 26 2025