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A383520 Heinz numbers of section-sum partitions with distinct multiplicities (Wilf).

%I A383520 #8 May 20 2025 23:43:53
%S A383520 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 A383520 44,45,47,49,50,52,53,56,59,61,64,67,68,71,73,75,76,79,80,81,83,88,89,
%U A383520 92,97,98,99,101,103,104,107,109,112,113,116,117,121,124,125
%N A383520 Heinz numbers of section-sum partitions with distinct multiplicities (Wilf).
%C A383520 First differs from A383515 in having 325.
%C A383520 First differs from A383532 in having 325.
%C A383520 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 A383520 An integer partition is Wilf iff its multiplicities are all different, ranked by A130091.
%C A383520 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.
%e A383520 The terms together with their prime indices begin:
%e A383520     1: {}
%e A383520     2: {1}
%e A383520     3: {2}
%e A383520     4: {1,1}
%e A383520     5: {3}
%e A383520     7: {4}
%e A383520     8: {1,1,1}
%e A383520     9: {2,2}
%e A383520    11: {5}
%e A383520    13: {6}
%e A383520    16: {1,1,1,1}
%e A383520    17: {7}
%e A383520    19: {8}
%e A383520    20: {1,1,3}
%e A383520    23: {9}
%e A383520    25: {3,3}
%e A383520    27: {2,2,2}
%e A383520    28: {1,1,4}
%e A383520    29: {10}
%e A383520    31: {11}
%e A383520    32: {1,1,1,1,1}
%t A383520 disjointFamilies[y_]:=Select[Tuples[IntegerPartitions/@Length/@Split[y]],UnsameQ@@Join@@#&];
%t A383520 prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A383520 conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
%t A383520 Select[Range[100],disjointFamilies[conj[prix[#]]]!={}&&UnsameQ@@Last/@FactorInteger[#]&]
%Y A383520 Ranking sequences are shown in parentheses below.
%Y A383520 For non Wilf instead of Wilf we have (A383514), counted by A383506.
%Y A383520 These partitions are counted by A383519.
%Y A383520 A055396 gives least prime index, greatest A061395.
%Y A383520 A056239 adds up prime indices, row sums of A112798, counted by A001222.
%Y A383520 A098859 counts Wilf partitions (A130091), conjugate (A383512).
%Y A383520 A122111 represents conjugation in terms of Heinz numbers.
%Y A383520 A239455 counts section-sum partitions (A381432), complement A351293 (A381433).
%Y A383520 A336866 counts non Wilf partitions (A130092), conjugate (A383513).
%Y A383520 A351592 counts non Wilf Look-and-Say partitions, ranked by (A384006).
%Y A383520 A381431 is the section-sum transform.
%Y A383520 Cf. A000720, A001223, A048767, A051903, A212166, A320348, A325366, A325368.
%K A383520 nonn
%O A383520 1,2
%A A383520 _Gus Wiseman_, May 19 2025