A322307 Number of multisets in the swell of the n-th multiset multisystem.

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}

Links

Table of n, a(n) for n=1..87.

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}]

Crossrefs

Cf. A003963, A054921, A056239, A112798, A218970, A286518, A290103, A301957, A302242, A304716, A305078, A305079, A316556, A322306.

Sequence in context: A293460 A231813 A158210 * A087802 A079553 A001221

Adjacent sequences: A322304 A322305 A322306 * A322308 A322309 A322310

Keyword

nonn

Author

Gus Wiseman, Dec 03 2018

Status

approved