A320923 Heinz numbers of connected graphical partitions.

4, 12, 27, 36, 40, 81, 90, 108, 112, 120, 225, 243, 252, 270, 300, 324, 336, 352, 360, 400, 567, 625, 630, 675, 729, 750, 756, 792, 810, 832, 840, 900, 972, 1000, 1008, 1056, 1080, 1120, 1200, 1323, 1575, 1701, 1750, 1764, 1782, 1872, 1875, 1890, 1980, 2025

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 graphical if it comprises the multiset of vertex-degrees of some connected simple graph.

Examples

			The sequence of all connected-graphical partitions begins: (11), (211), (222), (2211), (3111), (2222), (3221), (22211), (41111), (32111), (3322), (22222), (42211), (32221), (33211), (222211), (421111), (511111), (322111).
		

Crossrefs

Cf. A000070, A000569, A007717, A025065, A056239, A096373, A112798, A147878, A209816, A300061, A320635, A320911, A320921, A320922, A320925.

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}]]]]]]],And[UnsameQ@@#,Length[csm[#]]==1]&]!={}&]