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A383516 Heinz numbers of Look-and-Say partitions that are not section-sum partitions.

%I A383516 #6 May 19 2025 09:44:11
%S A383516 12,18,24,48,54,63,72,96,108,144,147,162,189,192,216,288,324,360,384,
%T A383516 432,486,504,540,567,576,600,648,720,756,768,792,864,936,972,1008,
%U A383516 1029,1152,1176,1188,1200,1224,1296,1323,1350,1368,1400,1404,1440,1458,1500
%N A383516 Heinz numbers of Look-and-Say partitions that are not section-sum partitions.
%C A383516 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 A383516 An integer partition is section-sum iff it is possible to choose a disjoint family of strict partitions, one of each of its positive 0-appended differences. These are ranked by A381432.
%C A383516 An integer partition is Look-and-Say iff it is possible to choose a disjoint family of strict partitions, one of each of its multiplicities. These are ranked by A351294.
%e A383516 The terms together with their prime indices begin:
%e A383516    12: {1,1,2}
%e A383516    18: {1,2,2}
%e A383516    24: {1,1,1,2}
%e A383516    48: {1,1,1,1,2}
%e A383516    54: {1,2,2,2}
%e A383516    63: {2,2,4}
%e A383516    72: {1,1,1,2,2}
%e A383516    96: {1,1,1,1,1,2}
%e A383516   108: {1,1,2,2,2}
%e A383516   144: {1,1,1,1,2,2}
%e A383516   147: {2,4,4}
%e A383516   162: {1,2,2,2,2}
%e A383516   189: {2,2,2,4}
%e A383516   192: {1,1,1,1,1,1,2}
%e A383516   216: {1,1,1,2,2,2}
%e A383516   288: {1,1,1,1,1,2,2}
%e A383516   324: {1,1,2,2,2,2}
%e A383516   360: {1,1,1,2,2,3}
%e A383516   384: {1,1,1,1,1,1,1,2}
%t A383516 disjointFamilies[y_]:=Select[Tuples[IntegerPartitions/@Length/@Split[y]],UnsameQ@@Join@@#&];
%t A383516 prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A383516 conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
%t A383516 Select[Range[100],disjointFamilies[prix[#]]!={}&&disjointFamilies[conj[prix[#]]]=={}&]
%Y A383516 Ranking sequences are shown in parentheses below.
%Y A383516 These partitions are counted by A383509.
%Y A383516 A048767 is the Look-and-Say transform, fixed points A048768, counted by A217605.
%Y A383516 A055396 gives least prime index, greatest A061395.
%Y A383516 A056239 adds up prime indices, row sums of A112798, counted by A001222.
%Y A383516 A098859 counts Wilf partitions (A130091), conjugate (A383512).
%Y A383516 A122111 represents conjugation in terms of Heinz numbers.
%Y A383516 A239455 counts Look-and-Say partitions (A351294), complement A351293 (A351295).
%Y A383516 A239455 counts section-sum partitions (A381432), complement A351293 (A381433).
%Y A383516 A336866 counts non Wilf partitions (A130092), conjugate (A383513).
%Y A383516 A381431 is the section-sum transform.
%Y A383516 A383508 counts partitions that are both Look-and-Say and section-sum (A383515).
%Y A383516 A383509 counts partitions that are not Look-and-Say but are section-sum (A384007).
%Y A383516 A383510 counts partitions that are neither Look-and-Say nor section-sum (A383517).
%Y A383516 A383511 counts partitions that are Look-and-Say and section-sum but not Wilf (A383518).
%Y A383516 Cf. A000720, A001223, A212166, A238745, A325368, A383514, A383520, A383531, A384006.
%K A383516 nonn
%O A383516 1,1
%A A383516 _Gus Wiseman_, May 18 2025