login

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).

A322307
Number of multisets in the swell of the n-th multiset multisystem.
6
0, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 2, 1, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 1, 2, 1, 3, 1, 1, 2, 2, 2, 2, 1, 2, 2, 2, 1, 3, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 3, 1, 2, 2, 1, 2, 3, 1, 2, 2, 3, 1, 2, 1, 2, 2, 2, 2, 3, 1, 2, 1, 2, 1, 3, 2, 2, 2
OFFSET
1,6
COMMENTS
First differs from A001221 at a(91) = 3, A001221(91) = 2.
The swell of a multiset partition is the set of possible joins of its connected submultisets, where the multiplicity of a vertex in the join of a set of multisets is the maximum multiplicity of the same vertex among the parts. For example the swell of {{1,1},{1,2},{2,2}} is:
{1,1}
{1,2}
{2,2}
{1,1,2}
{1,2,2}
{1,1,2,2}
MATHEMATICA
primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
zwell[y_]:=Union[y, Join@@Cases[Subsets[Union[y], {2}], {x_, z_}?(GCD@@#>1&):>zwell[Sort[Append[Fold[DeleteCases[#1, #2, {1}, 1]&, y, {x, z}], LCM[x, z]]]]]];
Table[Length[zwell[primeMS[n]]], {n, 100}]
KEYWORD
nonn
AUTHOR
Gus Wiseman, Dec 03 2018
STATUS
approved