OFFSET
1,5
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. 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}.
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.
EXAMPLE
The a(12) = 21 multisystems on {1,1,2,3} (commas elided):
{1123} {{1}{123}} {{1}{1}{23}} {{{1}}{{1}{23}}}
{{2}{113}} {{1}{2}{13}} {{{23}}{{1}{1}}}
{{3}{112}} {{1}{3}{12}} {{{1}}{{2}{13}}}
{{11}{23}} {{2}{3}{11}} {{{2}}{{1}{13}}}
{{12}{13}} {{{13}}{{1}{2}}}
{{{1}}{{3}{12}}}
{{{3}}{{1}{12}}}
{{{12}}{{1}{3}}}
{{{2}}{{3}{11}}}
{{{3}}{{2}{11}}}
{{{11}}{{2}{3}}}
MATHEMATICA
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]]]];
nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]], {#1}]&, If[n==1, {}, Flatten[Cases[FactorInteger[n]//Reverse, {p_, k_}:>Table[PrimePi[p], {k}]]]]];
tmsp[m_]:=Prepend[Join@@Table[tmsp[c], {c, Select[mps[m], 1<Length[#]<Length[m]&]}], m];
Table[Length[tmsp[nrmptn[n]]], {n, 20}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, Sep 04 2018
EXTENSIONS
Terminology corrected by Gus Wiseman, Jan 04 2020
More terms from Jinyuan Wang, Jun 26 2020
STATUS
approved