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A383518 Heinz numbers of integer partitions that are Look-and-Say and section-sum but not conjugate Wilf partitions.

%I A383518 #6 May 20 2025 21:58:42
%S A383518 325,845,931,1625,2527,3509,6253,6517,8125,9251
%N A383518 Heinz numbers of integer partitions that are Look-and-Say and section-sum but not conjugate Wilf partitions.
%C A383518 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 A383518 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.
%C A383518 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 A383518 A integer partition is Wilf iff its multiplicities are all different (ranked by A130091). It is conjugate Wilf iff its positive 0-appended differences are all different (ranked by A383512).
%e A383518 The terms together with their prime indices begin:
%e A383518    325: {3,3,6}
%e A383518    845: {3,6,6}
%e A383518    931: {4,4,8}
%e A383518   1625: {3,3,3,6}
%e A383518   2527: {4,8,8}
%e A383518   3509: {5,5,10}
%e A383518   6253: {6,6,12}
%e A383518   6517: {4,4,4,8}
%e A383518   8125: {3,3,3,3,6}
%e A383518   9251: {5,10,10}
%t A383518 disjointFamilies[y_]:=Select[Tuples[IntegerPartitions/@Length/@Split[y]],UnsameQ@@Join@@#&];
%t A383518 prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A383518 conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
%t A383518 Select[Range[1000],disjointFamilies[prix[#]]!={}&&disjointFamilies[conj[prix[#]]]!={}&&!UnsameQ@@Length/@Split[conj[prix[#]]]&]
%Y A383518 Ranking sequences are shown in parentheses below.
%Y A383518 These partitions are counted by A383511.
%Y A383518 A048767 is the Look-and-Say transform, fixed points A048768, counted by A217605.
%Y A383518 A055396 gives least prime index, greatest A061395.
%Y A383518 A056239 adds up prime indices, row sums of A112798, counted by A001222.
%Y A383518 A098859 counts Wilf partitions (A130091), conjugate (A383512).
%Y A383518 A122111 represents conjugation in terms of Heinz numbers.
%Y A383518 A239455 counts Look-and-Say partitions (A351294), complement A351293 (A351295).
%Y A383518 A239455 counts section-sum partitions (A381432), complement A351293 (A381433).
%Y A383518 A336866 counts non Wilf partitions (A130092), conjugate (A383513).
%Y A383518 A381431 is the section-sum transform.
%Y A383518 A383508 counts partitions that are both Look-and-Say and section-sum (A383515).
%Y A383518 A383509 counts partitions that are Look-and-Say but not section-sum (A383516).
%Y A383518 A383509 counts partitions that are not Look-and-Say but are section-sum (A384007).
%Y A383518 A383510 counts partitions that are neither Look-and-Say nor section-sum (A383517).
%Y A383518 Cf. A047966, A051903, A212166, A238745, A325368, A351592, A383531.
%K A383518 nonn,more
%O A383518 1,1
%A A383518 _Gus Wiseman_, May 18 2025