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A384006 Heinz numbers of Look-and-Say partitions without distinct multiplicities (non Wilf).

%I A384006 #5 May 22 2025 17:06:09
%S A384006 216,1000,1296,2744,3375,7776,9261,10000,10648,17576,32400,35937,
%T A384006 38416,38880,39304,42875,46656,50625,54000,54432,54872,59319,63504,
%U A384006 81000,85536,90000,97336,100000
%N A384006 Heinz numbers of Look-and-Say partitions without distinct multiplicities (non Wilf).
%C A384006 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 A384006 An integer partition is Wilf iff its multiplicities are all different, ranked by A130091, complement A130092.
%C A384006 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 A384006 The terms together with their prime indices begin:
%e A384006      216: {1,1,1,2,2,2}
%e A384006     1000: {1,1,1,3,3,3}
%e A384006     1296: {1,1,1,1,2,2,2,2}
%e A384006     2744: {1,1,1,4,4,4}
%e A384006     3375: {2,2,2,3,3,3}
%e A384006     7776: {1,1,1,1,1,2,2,2,2,2}
%e A384006     9261: {2,2,2,4,4,4}
%e A384006    10000: {1,1,1,1,3,3,3,3}
%e A384006    10648: {1,1,1,5,5,5}
%e A384006    17576: {1,1,1,6,6,6}
%e A384006    32400: {1,1,1,1,2,2,2,2,3,3}
%e A384006    35937: {2,2,2,5,5,5}
%e A384006    38416: {1,1,1,1,4,4,4,4}
%e A384006    38880: {1,1,1,1,1,2,2,2,2,2,3}
%e A384006    39304: {1,1,1,7,7,7}
%e A384006    42875: {3,3,3,4,4,4}
%e A384006    46656: {1,1,1,1,1,1,2,2,2,2,2,2}
%e A384006    50625: {2,2,2,2,3,3,3,3}
%e A384006    54000: {1,1,1,1,2,2,2,3,3,3}
%e A384006    54432: {1,1,1,1,1,2,2,2,2,2,4}
%e A384006    54872: {1,1,1,8,8,8}
%e A384006    59319: {2,2,2,6,6,6}
%e A384006    63504: {1,1,1,1,2,2,2,2,4,4}
%e A384006    81000: {1,1,1,2,2,2,2,3,3,3}
%e A384006    85536: {1,1,1,1,1,2,2,2,2,2,5}
%e A384006    90000: {1,1,1,1,2,2,3,3,3,3}
%e A384006    97336: {1,1,1,9,9,9}
%e A384006   100000: {1,1,1,1,1,3,3,3,3,3}
%t A384006 disjointFamilies[y_]:=Select[Tuples[IntegerPartitions/@Length/@Split[y]],UnsameQ@@Join@@#&];
%t A384006 prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A384006 Select[Range[1000],disjointFamilies[prix[#]]!={}&&!UnsameQ@@Last/@FactorInteger[#]&]
%Y A384006 Ranking sequences are shown in parentheses below.
%Y A384006 These partitions are counted by A351592.
%Y A384006 For section-sum instead of Look-and-Say we have (A383514), counted by A383506.
%Y A384006 A055396 gives least prime index, greatest A061395.
%Y A384006 A056239 adds up prime indices, row sums of A112798, counted by A001222.
%Y A384006 A098859 counts Wilf partitions (A130091), conjugate (A383512).
%Y A384006 A122111 represents conjugation in terms of Heinz numbers.
%Y A384006 A239455 counts section-sum partitions (A381432), complement A351293 (A381433).
%Y A384006 A336866 counts non Wilf partitions (A130092), conjugate (A383513).
%Y A384006 A383511 counts partitions that are Look-and-Say and section-sum but not Wilf (A383518).
%Y A384006 Cf. A000720, A001223, A048767, A051903, A212166, A325368, A383520, A383531.
%K A384006 nonn
%O A384006 1,1
%A A384006 _Gus Wiseman_, May 19 2025