%I #5 Oct 24 2018 19:21:33

%S 4,12,27,36,40,81,90,108,112,120,225,243,252,270,300,324,336,352,360,

%T 400,567,625,630,675,729,750,756,792,810,832,840,900,972,1000,1008,

%U 1056,1080,1120,1200,1323,1575,1701,1750,1764,1782,1872,1875,1890,1980,2025

%N Heinz numbers of connected graphical partitions.

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

%C An integer partition is connected and graphical if it comprises the multiset of vertex-degrees of some connected simple graph.

%e 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).

%t 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[[#]]&]}]]];

%t 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]]]]]]]]];

%t 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]&]!={}&]

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

%K nonn

%O 1,1

%A _Gus Wiseman_, Oct 24 2018