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A320924 revision #5

A320924

Heinz numbers of multigraphical partitions.

1, 4, 9, 12, 16, 25, 27, 30, 36, 40, 48, 49, 63, 64, 70, 75, 81, 84, 90, 100, 108, 112, 120, 121, 144, 147, 154, 160, 165, 169, 175, 189, 192, 196, 198, 210, 220, 225, 243, 250, 252, 256, 264, 270, 273, 280, 286, 289, 300, 324, 325, 336, 343, 351, 352, 360

(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 multigraphical if it comprises the multiset of vertex-degrees of some multigraph.

LINKS

Table of n, a(n) for n=1..56.

EXAMPLE

The sequence of all multigraphical partitions begins: (), (11), (22), (211), (1111), (33), (222), (321), (2211), (3111), (21111), (44), (422), (111111), (431), (332), (2222), (4211), (3221), (3311), (22211), (41111), (32111), (55), (221111).

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

CROSSREFS

Cf. A000070, A000569, A007717, A025065, A056239, A096373, A112798, A209816, A300061, A320459, A320911, A320922, A320925.

Sequence in context: A109424 A034019 A034018 * A357976 A330879 A357636

Adjacent sequences: A320921 A320922 A320923 * A320925 A320926 A320927

KEYWORD

nonn

AUTHOR

Gus Wiseman, Oct 24 2018

STATUS

proposed