%I #9 May 24 2021 00:13:29

%S 1,4,8,9,12,16,18,24,25,27,30,32,36,40,45,48,49,50,54,60,63,64,70,72,

%T 75,80,81,84,90,96,98,100,105,108,112,120,121,125,126,128,135,140,144,

%U 147,150,154,160,162,165,168,169,175,180,189,192,196,198,200,210

%N Heinz numbers of integer partitions that are the vertex-degrees of some set multipartition (multiset of nonempty sets) with no singletons.

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

%C Also Heinz numbers of partitions 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 at least twice the greatest prime index A061395(n). - _Gus Wiseman_, May 23 2021

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

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

%e 1: (): {}

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

%Y These partitions are counted by A110618.

%Y The even-weight version is A320924.

%Y The conjugate case of equality is A340387.

%Y The conjugate version is A344291.

%Y The opposite conjugate version is A344296.

%Y The opposite version is A344414.

%Y The case of equality is A344415.

%Y The opposite even-weight 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 A058696 counts partitions of even numbers, ranked by A300061.

%Y A334201 adds up all prime indices except the greatest.

%Y Cf. A000041, A000569, A265640, A283877, A306005, A318361, A320922, A320923, A320925.

%K nonn

%O 1,2

%A _Gus Wiseman_, Nov 26 2018