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Heinz numbers of graphical partitions.
37

%I #5 Oct 24 2018 19:21:26

%S 1,4,12,16,27,36,40,48,64,81,90,108,112,120,144,160,192,225,243,252,

%T 256,270,300,324,336,352,360,400,432,448,480,567,576,625,630,640,675,

%U 729,750,756,768,792,810,832,840,900,972,1000,1008,1024,1056,1080,1120

%N Heinz numbers of 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 graphical if it comprises the vertex-degrees of some simple graph.

%e The sequence of all graphical partitions begins: (), (11), (211), (1111), (222), (2211), (3111), (21111), (111111), (2222), (3221), (22211), (41111), (32111), (221111), (311111), (2111111), (3322), (22222), (42211).

%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 Select[Range[1000],Select[prptns[Flatten[MapIndexed[Table[#2,{#1}]&,If[#==1,{},Flatten[Cases[FactorInteger[#],{p_,k_}:>Table[PrimePi[p],{k}]]]]]]],UnsameQ@@#&]!={}&]

%Y Cf. A000070, A000569, A007717, A056239, A096373, A112798, A147878, A209816, A300061, A320458, A320911, A320923, A320924.

%K nonn

%O 1,2

%A _Gus Wiseman_, Oct 24 2018