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%I #12 May 24 2021 00:13:19
%S 1,4,9,12,16,25,27,30,36,40,48,49,63,64,70,75,81,84,90,100,108,112,
%T 120,121,144,147,154,160,165,169,175,189,192,196,198,210,220,225,243,
%U 250,252,256,264,270,273,280,286,289,300,324,325,336,343,351,352,360
%N Heinz numbers of multigraphical partitions.
%C The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).
%C An integer partition is multigraphical if it comprises the multiset of vertex-degrees of some multigraph.
%C 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
%F Members m of A300061 such that A061395(m) <= A056239(m)/2. - _Gus Wiseman_, May 23 2021
%e 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).
%e From _Gus Wiseman_, May 23 2021: (Start)
%e The sequence of terms together with their prime indices and a multigraph realizing each begins:
%e 1: () | {}
%e 4: (11) | {{1,2}}
%e 9: (22) | {{1,2},{1,2}}
%e 12: (112) | {{1,3},{2,3}}
%e 16: (1111) | {{1,2},{3,4}}
%e 25: (33) | {{1,2},{1,2},{1,2}}
%e 27: (222) | {{1,2},{1,3},{2,3}}
%e 30: (123) | {{1,3},{2,3},{2,3}}
%e 36: (1122) | {{1,2},{3,4},{3,4}}
%e 40: (1113) | {{1,4},{2,4},{3,4}}
%e 48: (11112) | {{1,2},{3,5},{4,5}}
%e 49: (44) | {{1,2},{1,2},{1,2},{1,2}}
%e 63: (224) | {{1,3},{1,3},{2,3},{2,3}}
%e (End)
%t 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[[#]]&]}]]];
%t Select[Range[1000],prptns[Flatten[MapIndexed[Table[#2,{#1}]&,If[#==1,{},Flatten[Cases[FactorInteger[#],{p_,k_}:>Table[PrimePi[p],{k}]]]]]]]!={}&]
%Y These partitions are counted by A209816.
%Y The case with odd weights is A322109.
%Y The conjugate case of equality is A340387.
%Y The conjugate version with odd weights allowed is A344291.
%Y The conjugate opposite version is A344292.
%Y The opposite version with odd weights allowed is A344296.
%Y The conjugate version is A344413.
%Y The conjugate opposite version with odd weights allowed is A344414.
%Y The case of equality is A344415.
%Y The opposite version is A344416.
%Y A000070 counts non-multigraphical partitions.
%Y A025065 counts palindromic partitions.
%Y A035363 counts partitions into even parts.
%Y A056239 adds up prime indices, row sums of A112798.
%Y A110618 counts partitions that are the vertex-degrees of some set multipartition with no singletons.
%Y A334201 adds up all prime indices except the greatest.
%Y Cf. A000041, A000569, A007717, A096373, A265640, A283877, A306005, A318361, A320459, A320911, A320922, A320923, A320925.
%K nonn
%O 1,2
%A _Gus Wiseman_, Oct 24 2018
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