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A383517 Heinz numbers of integer partitions that are neither Look-and-Say nor section-sum partitions.

%I A383517 #5 May 19 2025 09:44:03
%S A383517 6,21,30,36,42,60,66,70,78,84,90,102,105,110,114,120,126,132,138,140,
%T A383517 150,154,156,165,168,174,180,186,198,204,210,220,222,228,231,234,238,
%U A383517 240,246,252,258,264,270,273,276,280,282,286,294,300,306,308,312,315
%N A383517 Heinz numbers of integer partitions that are neither Look-and-Say nor section-sum partitions.
%C A383517 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 A383517 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, complement A381433.
%C A383517 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, complement A351295.
%e A383517 The terms together with their prime indices begin:
%e A383517     6: {1,2}
%e A383517    21: {2,4}
%e A383517    30: {1,2,3}
%e A383517    36: {1,1,2,2}
%e A383517    42: {1,2,4}
%e A383517    60: {1,1,2,3}
%e A383517    66: {1,2,5}
%e A383517    70: {1,3,4}
%e A383517    78: {1,2,6}
%e A383517    84: {1,1,2,4}
%e A383517    90: {1,2,2,3}
%e A383517   102: {1,2,7}
%e A383517   105: {2,3,4}
%e A383517   110: {1,3,5}
%e A383517   114: {1,2,8}
%e A383517   120: {1,1,1,2,3}
%t A383517 disjointFamilies[y_]:=Select[Tuples[IntegerPartitions/@Length/@Split[y]],UnsameQ@@Join@@#&];
%t A383517 prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A383517 conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
%t A383517 Select[Range[100],disjointFamilies[prix[#]]=={}&&disjointFamilies[conj[prix[#]]]=={}&]
%Y A383517 Ranking sequences are shown in parentheses below.
%Y A383517 These partitions are counted by A383510.
%Y A383517 A048767 is the Look-and-Say transform, fixed points A048768, counted by A217605.
%Y A383517 A055396 gives least prime index, greatest A061395.
%Y A383517 A056239 adds up prime indices, row sums of A112798, counted by A001222.
%Y A383517 A098859 counts Wilf partitions (A130091), conjugate (A383512).
%Y A383517 A122111 represents conjugation in terms of Heinz numbers.
%Y A383517 A239455 counts Look-and-Say partitions (A351294), complement A351293 (A351295).
%Y A383517 A239455 counts section-sum partitions (A381432), complement A351293 (A381433).
%Y A383517 A336866 counts non Wilf partitions (A130092), conjugate (A383513).
%Y A383517 A381431 is the section-sum transform.
%Y A383517 A383508 counts partitions that are both Look-and-Say and section-sum (A383515).
%Y A383517 A383509 counts partitions that are Look-and-Say but not section-sum (A383516).
%Y A383517 A383509 counts partitions that are not Look-and-Say but are section-sum (A384007).
%Y A383517 Cf. A000720, A001223, A051903, A212166, A238745, A325368, A383514, A383518, A383520, A384006.
%K A383517 nonn
%O A383517 1,1
%A A383517 _Gus Wiseman_, May 18 2025