|
|
A320922
|
|
Heinz numbers of graphical partitions.
|
|
37
|
|
|
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
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,2
|
|
COMMENTS
|
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.
|
|
LINKS
|
|
|
EXAMPLE
|
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).
|
|
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[[#]]&]}]]];
Select[Range[1000], Select[prptns[Flatten[MapIndexed[Table[#2, {#1}]&, If[#==1, {}, Flatten[Cases[FactorInteger[#], {p_, k_}:>Table[PrimePi[p], {k}]]]]]]], UnsameQ@@#&]!={}&]
|
|
CROSSREFS
|
Cf. A000070, A000569, A007717, A056239, A096373, A112798, A147878, A209816, A300061, A320458, A320911, A320923, A320924.
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|