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

1, 1, 1, 1, 1, 2, 2, 3, 7, 5, 5, 11, 16, 16, 27, 18, 61, 62, 272, 45, 123, 61, 1385, 105, 152, 272, 501, 211, 7936, 362

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}.

Links

Table of n, a(n) for n=1..30.

Formula

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

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

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

The version with distinct atoms is A006472.

The non-maximal version is A318846.

A tree version is A318848, with orderless version A318849.

The unlabeled version is A330664.

Final terms in each row of A330727.

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

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

Sequence in context: A354950 A267822 A210598 * A354377 A051301 A002583

Adjacent sequences: A330725 A330726 A330727 * A330729 A330730 A330731

Keyword

nonn,more

Author

Gus Wiseman, Dec 30 2019

Status

approved