OFFSET
1,6
COMMENTS
A balanced reduced multisystem is either a finite multiset, or a multiset partition with at least two parts, not all of which are singletons, of a balanced reduced multisystem.
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. A multiset whose multiplicities are the prime indices of n (such as 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}.
EXAMPLE
The a(n) multisystems for n = 3, 6, 8, 9, 10, 12 (commas and outer brackets elided):
11 {1}{12} {1}{23} {{1}}{{1}{22}} {{1}}{{1}{12}} {{1}}{{1}{23}}
{2}{11} {2}{13} {{11}}{{2}{2}} {{11}}{{1}{2}} {{11}}{{2}{3}}
{3}{12} {{1}}{{2}{12}} {{1}}{{2}{11}} {{1}}{{2}{13}}
{{12}}{{1}{2}} {{12}}{{1}{1}} {{12}}{{1}{3}}
{{2}}{{1}{12}} {{2}}{{1}{11}} {{1}}{{3}{12}}
{{2}}{{2}{11}} {{13}}{{1}{2}}
{{22}}{{1}{1}} {{2}}{{1}{13}}
{{2}}{{3}{11}}
{{23}}{{1}{1}}
{{3}}{{1}{12}}
{{3}}{{2}{11}}
MATHEMATICA
nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]], {#1}]&, If[n==1, {}, Flatten[Cases[Reverse[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]]]];
totm[m_]:=Prepend[Join@@Table[totm[p], {p, Select[mps[m], 1<Length[#]<Length[m]&]}], m];
Table[Length[Select[totm[nrmptn[n]], Depth[#]==If[n<=2, 2, Length[nrmptn[n]]]&]], {n, 20}]
CROSSREFS
KEYWORD
nonn,more
AUTHOR
Gus Wiseman, Dec 30 2019
STATUS
approved