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A384010 Heinz numbers of integer partitions such that it is possible to choose a family of disjoint strict partitions, one of each conjugate part.

%I A384010 #10 May 24 2025 11:00:35
%S A384010 1,2,4,6,8,12,16,18,24,27,30,32,36,48,54,60,64,72,81,90,96,108,120,
%T A384010 128,144,150,162,180,192
%N A384010 Heinz numbers of integer partitions such that it is possible to choose a family of disjoint strict partitions, one of each conjugate part.
%C A384010 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.
%e A384010 The prime indices of 96 are {1,1,1,1,1,2}, conjugate (6,1), disjoint family (4,2,1), so 96 is in the sequence.
%e A384010 The terms together with their prime indices begin:
%e A384010     1: {}
%e A384010     2: {1}
%e A384010     4: {1,1}
%e A384010     6: {1,2}
%e A384010     8: {1,1,1}
%e A384010    12: {1,1,2}
%e A384010    16: {1,1,1,1}
%e A384010    18: {1,2,2}
%e A384010    24: {1,1,1,2}
%e A384010    27: {2,2,2}
%e A384010    30: {1,2,3}
%e A384010    32: {1,1,1,1,1}
%e A384010    36: {1,1,2,2}
%e A384010    48: {1,1,1,1,2}
%e A384010    54: {1,2,2,2}
%e A384010    60: {1,1,2,3}
%e A384010    64: {1,1,1,1,1,1}
%t A384010 pof[y_]:=Select[Join@@@Tuples[IntegerPartitions/@y],UnsameQ@@#&];
%t A384010 conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
%t A384010 prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A384010 Select[Range[100],pof[conj[prix[#]]]!={}&]
%Y A384010 For multiplicities instead of indices we have A382525.
%Y A384010 These partitions are counted by A383708, without ones A383533, complement A383711.
%Y A384010 These are the positions of positive terms in A384005.
%Y A384010 The complement is A384011, conjugate A383710.
%Y A384010 A000041 counts integer partitions, strict A000009.
%Y A384010 A048767 is the Look-and-Say transform, fixed points A048768, counted by A217605.
%Y A384010 A055396 gives least prime index, greatest A061395.
%Y A384010 A056239 adds up prime indices, row sums of A112798.
%Y A384010 A122111 represent conjugation in terms of Heinz numbers.
%Y A384010 A239455 counts Look-and-Say or section-sum partitions, ranks A351294 or A381432.
%Y A384010 A351293 counts non-Look-and-Say or non-section-sum partitions, ranks A351295 or A381433.
%Y A384010 Cf. A098859, A130091, A279375, A279790, A299200, A357982, A382912, A383706, A383707.
%K A384010 nonn,more
%O A384010 1,2
%A A384010 _Gus Wiseman_, May 23 2025