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A320923
Heinz numbers of connected graphical partitions.
19
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
1,1
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.
EXAMPLE
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).
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]&]!={}&]
KEYWORD
nonn
AUTHOR
Gus Wiseman, Oct 24 2018
STATUS
approved