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A383515 Heinz numbers of integer partitions that are both Look-and-Say and section-sum.

%I A383515 #5 May 19 2025 09:44:07
%S A383515 1,2,3,4,5,7,8,9,11,13,16,17,19,20,23,25,27,28,29,31,32,37,40,41,43,
%T A383515 44,45,47,49,50,52,53,56,59,61,64,67,68,71,73,75,76,79,80,81,83,88,89,
%U A383515 92,97,98,99,101,103,104,107,109,112,113,116,117,121,124,125
%N A383515 Heinz numbers of integer partitions that are both Look-and-Say and section-sum.
%C A383515 First differs from A383532 in having 325.
%C A383515 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 A383515 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 A383515 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 A383515 The terms together with their prime indices begin:
%e A383515    1: {}
%e A383515    2: {1}
%e A383515    3: {2}
%e A383515    4: {1,1}
%e A383515    5: {3}
%e A383515    7: {4}
%e A383515    8: {1,1,1}
%e A383515    9: {2,2}
%e A383515   11: {5}
%e A383515   13: {6}
%e A383515   16: {1,1,1,1}
%e A383515   17: {7}
%e A383515   19: {8}
%e A383515   20: {1,1,3}
%e A383515   23: {9}
%e A383515   25: {3,3}
%e A383515   27: {2,2,2}
%e A383515   28: {1,1,4}
%e A383515   29: {10}
%e A383515   31: {11}
%e A383515   32: {1,1,1,1,1}
%t A383515 disjointFamilies[y_]:=Select[Tuples[IntegerPartitions/@Length/@Split[y]],UnsameQ@@Join@@#&];
%t A383515 prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A383515 conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
%t A383515 Select[Range[100],disjointFamilies[prix[#]]!={}&&disjointFamilies[conj[prix[#]]]!={}&]
%Y A383515 Ranking sequences are shown in parentheses below.
%Y A383515 These partitions are counted by A383508.
%Y A383515 A048767 is the Look-and-Say transform.
%Y A383515 A048768 gives Look-and-Say fixed points, counted by A217605.
%Y A383515 A055396 gives least prime index, greatest A061395.
%Y A383515 A056239 adds up prime indices, row sums of A112798, counted by A001222.
%Y A383515 A098859 counts Wilf partitions (A130091), conjugate (A383512).
%Y A383515 A122111 represents conjugation in terms of Heinz numbers.
%Y A383515 A239455 counts Look-and-Say partitions (A351294), complement A351293 (A351295).
%Y A383515 A239455 counts section-sum partitions (A381432), complement A351293 (A381433).
%Y A383515 A336866 counts non Wilf partitions (A130092), conjugate (A383513).
%Y A383515 A381431 is the section-sum transform.
%Y A383515 A383509 counts partitions that are Look-and-Say but not section-sum (A383516).
%Y A383515 A383509 counts partitions that are not Look-and-Say but are section-sum (A384007).
%Y A383515 A383510 counts partitions that are neither Look-and-Say nor section-sum (A383517).
%Y A383515 A383511 counts partitions that are Look-and-Say and section-sum but not Wilf (A383518).
%Y A383515 Cf. A000720, A001223, A047966, A051903, A109298, A212166, A238745, A325368, A351592, A384006.
%K A383515 nonn
%O A383515 1,2
%A A383515 _Gus Wiseman_, May 18 2025