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%I #15 Jun 26 2020 11:55:45
%S 1,1,1,1,2,3,6,4,15,11,20,21,90,51,80,32,468,166,2910,124,521,277,
%T 20644,266,621,1761,1866,841,165874,1374,1484344,436,3797,12741,5383,
%U 3108,14653890,103783,31323,2294,158136988,12419,1852077284,6382,20786,939131,23394406084
%N Number of balanced reduced multisystems whose atoms cover an initial interval of positive integers with multiplicities equal to the prime indices of n.
%C 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}.
%C 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.
%F a(n) = A318812(A181821(n)).
%F a(prime(n)) = A318813(n).
%F a(2^n) = A005121(n).
%e The a(12) = 21 multisystems on {1,1,2,3} (commas elided):
%e {1123} {{1}{123}} {{1}{1}{23}} {{{1}}{{1}{23}}}
%e {{2}{113}} {{1}{2}{13}} {{{23}}{{1}{1}}}
%e {{3}{112}} {{1}{3}{12}} {{{1}}{{2}{13}}}
%e {{11}{23}} {{2}{3}{11}} {{{2}}{{1}{13}}}
%e {{12}{13}} {{{13}}{{1}{2}}}
%e {{{1}}{{3}{12}}}
%e {{{3}}{{1}{12}}}
%e {{{12}}{{1}{3}}}
%e {{{2}}{{3}{11}}}
%e {{{3}}{{2}{11}}}
%e {{{11}}{{2}{3}}}
%t sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
%t nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,If[n==1,{},Flatten[Cases[FactorInteger[n]//Reverse,{p_,k_}:>Table[PrimePi[p],{k}]]]]];
%t tmsp[m_]:=Prepend[Join@@Table[tmsp[c],{c,Select[mps[m],1<Length[#]<Length[m]&]}],m];
%t Table[Length[tmsp[nrmptn[n]]],{n,20}]
%Y Row sums of A330727.
%Y Cf. A001055, A002846, A005121, A181821, A213427, A281118, A281119, A305936, A318762, A318812, A318813.
%Y Cf. A318847, A318848, A318849.
%K nonn
%O 1,5
%A _Gus Wiseman_, Sep 04 2018
%E Terminology corrected by _Gus Wiseman_, Jan 04 2020
%E More terms from _Jinyuan Wang_, Jun 26 2020
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