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A384320 Heinz numbers of integer partitions whose distinct parts are maximally refined.

%I A384320 #16 Jun 10 2025 16:26:02
%S A384320 1,2,3,4,6,8,9,10,12,14,15,16,18,20,24,27,28,30,32,36,40,42,45,48,50,
%T A384320 54,56,60,64,66,70,72,75,78,80,81,84,90,96,98,100,105,108,110,112,120,
%U A384320 126,128,132,135,140,144,150,156,160,162,168,180,182,192,196
%N A384320 Heinz numbers of integer partitions whose distinct parts are maximally refined.
%C A384320 The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.
%C A384320 Given a partition, the following are equivalent:
%C A384320 1) The distinct parts are maximally refined.
%C A384320 2) Every strict partition of a part contains a part. In other words, if y is the set of parts and z is any strict partition of any element of y, then z must contain at least one element from y.
%C A384320 3) No part is a sum of distinct non-parts.
%e A384320 The terms together with their prime indices begin:
%e A384320     1: {}
%e A384320     2: {1}
%e A384320     3: {2}
%e A384320     4: {1,1}
%e A384320     6: {1,2}
%e A384320     8: {1,1,1}
%e A384320     9: {2,2}
%e A384320    10: {1,3}
%e A384320    12: {1,1,2}
%e A384320    14: {1,4}
%e A384320    15: {2,3}
%e A384320    16: {1,1,1,1}
%e A384320    18: {1,2,2}
%e A384320    20: {1,1,3}
%e A384320    24: {1,1,1,2}
%e A384320    27: {2,2,2}
%e A384320    28: {1,1,4}
%e A384320    30: {1,2,3}
%e A384320    32: {1,1,1,1,1}
%t A384320 prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A384320 nonsets[y_]:=If[Length[y]==0,{},Rest[Subsets[Complement[Range[Max@@y],y]]]];
%t A384320 Select[Range[20],With[{y=Union[prix[#]]},UnsameQ@@y&&Intersection[y,Total/@nonsets[y]]=={}]&]
%Y A384320 The squarefree case is A383707, counted by A179009.
%Y A384320 The complement appears to be A384321, strict case A384322, counted by A384318.
%Y A384320 Partitions of this type are counted by A384392.
%Y A384320 A048767 is the Look-and-Say transform, fixed points A048768.
%Y A384320 A055396 gives least prime index, greatest A061395.
%Y A384320 A056239 adds up prime indices, row sums of A112798.
%Y A384320 Cf. A383706, A357982 (non-disjoint), A299200 (non-strict).
%Y A384320 Cf. A130091, A279375, A279790, A317142, A326080, A351294, A351295, A381454, A382525, A384390.
%K A384320 nonn
%O A384320 1,2
%A A384320 _Gus Wiseman_, Jun 01 2025