%I #4 Dec 31 2019 08:23:43

%S 1,1,1,1,1,2,2,3,7,5,5,11,16,16,27,18,61,62,272,45,123,61,1385,105,

%T 152,272,501,211,7936,362

%N Number of balanced reduced multisystems of maximum depth whose degrees (atom multiplicities) are the prime indices of n.

%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.

%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}.

%F a(2^n) = A006472(n).

%F a(prime(n)) = A000111(n - 1).

%e The a(n) multisystems for n = 3, 6, 8, 9, 10, 12 (commas and outer brackets elided):

%e 11 {1}{12} {1}{23} {{1}}{{1}{22}} {{1}}{{1}{12}} {{1}}{{1}{23}}

%e {2}{11} {2}{13} {{11}}{{2}{2}} {{11}}{{1}{2}} {{11}}{{2}{3}}

%e {3}{12} {{1}}{{2}{12}} {{1}}{{2}{11}} {{1}}{{2}{13}}

%e {{12}}{{1}{2}} {{12}}{{1}{1}} {{12}}{{1}{3}}

%e {{2}}{{1}{12}} {{2}}{{1}{11}} {{1}}{{3}{12}}

%e {{2}}{{2}{11}} {{13}}{{1}{2}}

%e {{22}}{{1}{1}} {{2}}{{1}{13}}

%e {{2}}{{3}{11}}

%e {{23}}{{1}{1}}

%e {{3}}{{1}{12}}

%e {{3}}{{2}{11}}

%t nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,If[n==1,{},Flatten[Cases[Reverse[FactorInteger[n]],{p_,k_}:>Table[PrimePi[p],{k}]]]]];

%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 totm[m_]:=Prepend[Join@@Table[totm[p],{p,Select[mps[m],1<Length[#]<Length[m]&]}],m];

%t Table[Length[Select[totm[nrmptn[n]],Depth[#]==If[n<=2,2,Length[nrmptn[n]]]&]],{n,20}]

%Y The version with distinct atoms is A006472.

%Y The non-maximal version is A318846.

%Y A tree version is A318848, with orderless version A318849.

%Y The unlabeled version is A330664.

%Y Final terms in each row of A330727.

%Y See also A330675 (strongly normal), A330676 (normal), and A330726 (partition).

%Y Cf. A000111, A001055, A005121, A292504, A292505, A317145, A330665, A330666.

%K nonn,more

%O 1,6

%A _Gus Wiseman_, Dec 30 2019