proposed

approved

editing

proposed

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

allocated for Gus WisemanHeinz numbers of graphical partitions.

1, 4, 12, 16, 27, 36, 40, 48, 64, 81, 90, 108, 112, 120, 144, 160, 192, 225, 243, 252, 256, 270, 300, 324, 336, 352, 360, 400, 432, 448, 480, 567, 576, 625, 630, 640, 675, 729, 750, 756, 768, 792, 810, 832, 840, 900, 972, 1000, 1008, 1024, 1056, 1080, 1120

1,2

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

An integer partition is graphical if it comprises the vertex-degrees of some simple graph.

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).

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[[#]]&]}]]];

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

Cf. A000070, A000569, A007717, A056239, A096373, A112798, A147878, A209816, A300061, A320458, A320911.

allocated

nonn

Gus Wiseman, Oct 24 2018

approved

editing

allocated for Gus Wiseman

allocated

approved