A320925 Heinz numbers of connected multigraphical partitions.

4, 9, 12, 25, 27, 30, 36, 40, 49, 63, 70, 75, 81, 84, 90, 100, 108, 112, 120, 121, 147, 154, 165, 169, 175, 189, 196, 198, 210, 220, 225, 243, 250, 252, 264, 270, 273, 280, 286, 289, 300, 324, 325, 336, 343, 351, 352, 360, 361, 363, 364, 385, 390, 400, 441

Offset: 1

Author

Gus Wiseman, Oct 24 2018

Keywords

• nonn

Comments

The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

An integer partition is connected and multigraphical if it comprises the multiset of vertex-degrees of some connected multigraph.

Examples

			The sequence of all connected multigraphical partitions begins: (11), (22), (211), (33), (222), (321), (2211), (3111), (44), (422), (431), (332), (2222), (4211), (3221), (3311), (22211), (41111), (32111).
		

Crossrefs

Cf. A000070, A000569, A007717, A056239, A112798, A147878, A209816, A300061, A320459, A320911, A320921, A320923, A320924.

Programs

Mathematica

prptns[m_]:=Union[Sort/@If[Length[m]==0,{{}},Join@@Table[Prepend[#,m[[ipr]]]&/@prptns[Delete[m,List/@ipr]],{ipr,Select[Prepend[{#},1]&/@Select[Range[2,Length[m]],m[[#]]>m[[#-1]]&],UnsameQ@@m[[#]]&]}]]];
csm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[OrderedQ[#],UnsameQ@@#,Length[Intersection@@s[[#]]]>0]&]},If[c=={},s,csm[Union[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
Select[Range[1000],Select[prptns[Flatten[MapIndexed[Table[#2,{#1}]&,If[#==1,{},Flatten[Cases[FactorInteger[#],{p_,k_}:>Table[PrimePi[p],{k}]]]]]]],Length[csm[#]]==1&]!={}&]