cp's OEIS Frontend

A322109 Heinz numbers of integer partitions that are the vertex-degrees of some set multipartition (multiset of nonempty sets) with no singletons.

%I A322109 #9 May 24 2021 00:13:29
%S A322109 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 A322109 75,80,81,84,90,96,98,100,105,108,112,120,121,125,126,128,135,140,144,
%U A322109 147,150,154,160,162,165,168,169,175,180,189,192,196,198,200,210
%N A322109 Heinz numbers of integer partitions that are the vertex-degrees of some set multipartition (multiset of nonempty sets) with no singletons.
%C A322109 The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).
%C A322109 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 A322109 A061395(a(n)) <= A056239(a(n))/2.
%e A322109 Each term paired with its Heinz partition and a realizing set multipartition with no singletons:
%e A322109    1:      (): {}
%e A322109    4:    (11): {{1,2}}
%e A322109    8:   (111): {{1,2,3}}
%e A322109    9:    (22): {{1,2},{1,2}}
%e A322109   12:   (211): {{1,2},{1,3}}
%e A322109   16:  (1111): {{1,2,3,4}}
%e A322109   18:   (221): {{1,2},{1,2,3}}
%e A322109   24:  (2111): {{1,2},{1,3,4}}
%e A322109   25:    (33): {{1,2},{1,2},{1,2}}
%e A322109   27:   (222): {{1,2,3},{1,2,3}}
%e A322109   30:   (321): {{1,2},{1,2},{1,3}}
%e A322109   32: (11111): {{1,2,3,4,5}}
%e A322109   36:  (2211): {{1,2},{1,2,3,4}}
%e A322109   40:  (3111): {{1,2},{1,3},{1,4}}
%t A322109 nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,If[n==1,{},Flatten[Cases[FactorInteger[n]//Reverse,{p_,k_}:>Table[PrimePi[p],{k}]]]]];
%t A322109 sqnopfacs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[sqnopfacs[n/d],Min@@#>=d&]],{d,Select[Rest[Divisors[n]],!PrimeQ[#]&&SquareFreeQ[#]&]}]]
%t A322109 Select[Range[100],Length[sqnopfacs[Times@@Prime/@nrmptn[#]]]>0&]
%Y A322109 These partitions are counted by A110618.
%Y A322109 The even-weight version is A320924.
%Y A322109 The conjugate case of equality is A340387.
%Y A322109 The conjugate version is A344291.
%Y A322109 The opposite conjugate version is A344296.
%Y A322109 The opposite version is A344414.
%Y A322109 The case of equality is A344415.
%Y A322109 The opposite even-weight version is A344416.
%Y A322109 A000070 counts non-multigraphical partitions.
%Y A322109 A025065 counts palindromic partitions.
%Y A322109 A035363 counts partitions into even parts.
%Y A322109 A056239 adds up prime indices, row sums of A112798.
%Y A322109 A058696 counts partitions of even numbers, ranked by A300061.
%Y A322109 A334201 adds up all prime indices except the greatest.
%Y A322109 Cf. A000041, A000569, A265640, A283877, A306005, A318361, A320922, A320923, A320925.
%K A322109 nonn
%O A322109 1,2
%A A322109 _Gus Wiseman_, Nov 26 2018