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The conjugate case of equality is A340387, counted by A035363.

The conjugate version is A344291, also counted by A110618.

The opposite conjugate version is A344296, counted by A025065.

The opposite version is A344414, also counted by A025065.

The case of equality is A344415, also counted by A035363.

The opposite even-weight version is A344416, counted by A000070.

A000070 counts non-multigraphical partitions.

A025065 counts palindromic partitions.

A035363 counts partitions into even parts.

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

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

A061395(a(n)) <= A056239(a(n))/2.

These partitions are counted by A110618.

The even-weight version is A320924.

The conjugate case of equality is A340387, counted by A035363.

The conjugate version is A344291, also counted by A110618.

The opposite conjugate version is A344296, counted by A025065.

The opposite version is A344414, also counted by A025065.

The case of equality is A344415, also counted by A035363.

The opposite even-weight version is A344416, counted by A000070.

A056239 adds up prime indices, row sums of A112798.

A058696 counts partitions of even numbers, ranked by A300061.

A334201 adds up all prime indices except the greatest.

Cf. A000070, A000041, A000569, A056239, A110618, A112798, A265640, A283877, A306005, A318361, A320922, A320923, A320924, A320925.

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allocated for Gus WisemanHeinz numbers of integer partitions that are the vertex-degrees of some set multipartition (multiset of nonempty sets) with no singletons.

1, 4, 8, 9, 12, 16, 18, 24, 25, 27, 30, 32, 36, 40, 45, 48, 49, 50, 54, 60, 63, 64, 70, 72, 75, 80, 81, 84, 90, 96, 98, 100, 105, 108, 112, 120, 121, 125, 126, 128, 135, 140, 144, 147, 150, 154, 160, 162, 165, 168, 169, 175, 180, 189, 192, 196, 198, 200, 210

1,2

The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

Each term paired with its Heinz partition and a realizing set multipartition with no singletons:

1: (): {}

4: (11): {{1,2}}

8: (111): {{1,2,3}}

9: (22): {{1,2},{1,2}}

12: (211): {{1,2},{1,3}}

16: (1111): {{1,2,3,4}}

18: (221): {{1,2},{1,2,3}}

24: (2111): {{1,2},{1,3,4}}

25: (33): {{1,2},{1,2},{1,2}}

27: (222): {{1,2,3},{1,2,3}}

30: (321): {{1,2},{1,2},{1,3}}

32: (11111): {{1,2,3,4,5}}

36: (2211): {{1,2},{1,2,3,4}}

40: (3111): {{1,2},{1,3},{1,4}}

nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]], {#1}]&, If[n==1, {}, Flatten[Cases[FactorInteger[n]//Reverse, {p_, k_}:>Table[PrimePi[p], {k}]]]]];

sqnopfacs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[sqnopfacs[n/d], Min@@#>=d&]], {d, Select[Rest[Divisors[n]], !PrimeQ[#]&&SquareFreeQ[#]&]}]]

Select[Range[100], Length[sqnopfacs[Times@@Prime/@nrmptn[#]]]>0&]

Cf. A000070, A000569, A056239, A110618, A112798, A283877, A306005, A318361, A320922, A320923, A320924, A320925.

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Gus Wiseman, Nov 26 2018

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allocated for Gus Wiseman

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