cp's OEIS Frontend

A330728 Number of balanced reduced multisystems of maximum depth whose degrees (atom multiplicities) are the prime indices of n.

%I A330728 #4 Dec 31 2019 08:23:43
%S A330728 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 A330728 152,272,501,211,7936,362
%N A330728 Number of balanced reduced multisystems of maximum depth whose degrees (atom multiplicities) are the prime indices of n.
%C A330728 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 A330728 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 A330728 a(2^n) = A006472(n).
%F A330728 a(prime(n)) = A000111(n - 1).
%e A330728 The a(n) multisystems for n = 3, 6, 8, 9, 10, 12 (commas and outer brackets elided):
%e A330728   11  {1}{12}  {1}{23}  {{1}}{{1}{22}}  {{1}}{{1}{12}}  {{1}}{{1}{23}}
%e A330728       {2}{11}  {2}{13}  {{11}}{{2}{2}}  {{11}}{{1}{2}}  {{11}}{{2}{3}}
%e A330728                {3}{12}  {{1}}{{2}{12}}  {{1}}{{2}{11}}  {{1}}{{2}{13}}
%e A330728                         {{12}}{{1}{2}}  {{12}}{{1}{1}}  {{12}}{{1}{3}}
%e A330728                         {{2}}{{1}{12}}  {{2}}{{1}{11}}  {{1}}{{3}{12}}
%e A330728                         {{2}}{{2}{11}}                  {{13}}{{1}{2}}
%e A330728                         {{22}}{{1}{1}}                  {{2}}{{1}{13}}
%e A330728                                                         {{2}}{{3}{11}}
%e A330728                                                         {{23}}{{1}{1}}
%e A330728                                                         {{3}}{{1}{12}}
%e A330728                                                         {{3}}{{2}{11}}
%t A330728 nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,If[n==1,{},Flatten[Cases[Reverse[FactorInteger[n]],{p_,k_}:>Table[PrimePi[p],{k}]]]]];
%t A330728 sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t A330728 mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
%t A330728 totm[m_]:=Prepend[Join@@Table[totm[p],{p,Select[mps[m],1<Length[#]<Length[m]&]}],m];
%t A330728 Table[Length[Select[totm[nrmptn[n]],Depth[#]==If[n<=2,2,Length[nrmptn[n]]]&]],{n,20}]
%Y A330728 The version with distinct atoms is A006472.
%Y A330728 The non-maximal version is A318846.
%Y A330728 A tree version is A318848, with orderless version A318849.
%Y A330728 The unlabeled version is A330664.
%Y A330728 Final terms in each row of A330727.
%Y A330728 See also A330675 (strongly normal), A330676 (normal), and A330726 (partition).
%Y A330728 Cf. A000111, A001055, A005121, A292504, A292505, A317145, A330665, A330666.
%K A330728 nonn,more
%O A330728 1,6
%A A330728 _Gus Wiseman_, Dec 30 2019