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

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.

Also Heinz numbers of integer partitions of even numbers whose greatest part is less than or equal to half the sum of parts, i.e., numbers n whose sum of prime indices A056239(n) is even and at least twice the greatest prime index A061395(n). - Gus Wiseman, May 23 2021

Links

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

Formula

Members m of A300061 such that A061395(m) <= A056239(m)/2. - Gus Wiseman, May 23 2021

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).
From Gus Wiseman, May 23 2021: (Start)
The sequence of terms together with their prime indices and a multigraph realizing each begins:
    1:      () | {}
    4:    (11) | {{1,2}}
    9:    (22) | {{1,2},{1,2}}
   12:   (112) | {{1,3},{2,3}}
   16:  (1111) | {{1,2},{3,4}}
   25:    (33) | {{1,2},{1,2},{1,2}}
   27:   (222) | {{1,2},{1,3},{2,3}}
   30:   (123) | {{1,3},{2,3},{2,3}}
   36:  (1122) | {{1,2},{3,4},{3,4}}
   40:  (1113) | {{1,4},{2,4},{3,4}}
   48: (11112) | {{1,2},{3,5},{4,5}}
   49:    (44) | {{1,2},{1,2},{1,2},{1,2}}
   63:   (224) | {{1,3},{1,3},{2,3},{2,3}}
(End)

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

These partitions are counted by A209816.

The case with odd weights is A322109.

The conjugate case of equality is A340387.

The conjugate version with odd weights allowed is A344291.

The conjugate opposite version is A344292.

The opposite version with odd weights allowed is A344296.

The conjugate version is A344413.

The conjugate opposite version with odd weights allowed is A344414.

The case of equality is A344415.

The opposite version is A344416.

A000070 counts non-multigraphical partitions.

A025065 counts palindromic partitions.

A035363 counts partitions into even parts.

A056239 adds up prime indices, row sums of A112798.

A110618 counts partitions that are the vertex-degrees of some set multipartition with no singletons.

A334201 adds up all prime indices except the greatest.

Cf. A000041, A000569, A007717, A096373, A265640, A283877, A306005, A318361, A320459, A320911, A320922, A320923, 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

approved