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A383532 Heinz numbers of integer partitions with distinct multiplicities (Wilf) and distinct nonzero 0-appended differences (conjugate Wilf).

%I A383532 #9 May 16 2025 18:50:47
%S A383532 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 A383532 44,45,47,49,50,52,53,56,59,61,64,67,68,71,73,75,76,79,80,81,83,88,89,
%U A383532 92,97,98,99,101,103,104,107,109,112,113,116,117,121,124,125
%N A383532 Heinz numbers of integer partitions with distinct multiplicities (Wilf) and distinct nonzero 0-appended differences (conjugate Wilf).
%C A383532 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 A383532 An integer partition is Wilf iff its multiplicities are all different (ranked by A130091). It is conjugate Wilf iff its nonzero 0-appended differences are all different (ranked by A383512).
%F A383532 Equals A130091 /\ A383512.
%e A383532 The terms together with their prime indices begin:
%e A383532     1: {}
%e A383532     2: {1}
%e A383532     3: {2}
%e A383532     4: {1,1}
%e A383532     5: {3}
%e A383532     7: {4}
%e A383532     8: {1,1,1}
%e A383532     9: {2,2}
%e A383532    11: {5}
%e A383532    13: {6}
%e A383532    16: {1,1,1,1}
%e A383532    17: {7}
%e A383532    19: {8}
%e A383532    20: {1,1,3}
%e A383532    23: {9}
%e A383532    25: {3,3}
%e A383532    27: {2,2,2}
%e A383532    28: {1,1,4}
%e A383532    29: {10}
%e A383532    31: {11}
%e A383532    32: {1,1,1,1,1}
%t A383532 prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n], {p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A383532 paug[y_]:=-DeleteCases[Differences[Append[y,0]],0];
%t A383532 Select[Range[100], UnsameQ@@Last/@FactorInteger[#] && UnsameQ@@paug[Reverse[prix[#]]]&]
%Y A383532 Partitions of this type are counted by A383507.
%Y A383532 Negating both sides gives A383531, counted by A383530.
%Y A383532 A048767 is the Look-and-Say transform, union A351294, complement A351295.
%Y A383532 A055396 gives least prime index, greatest A061395.
%Y A383532 A056239 adds up prime indices, row sums of A112798, counted by A001222.
%Y A383532 A098859 counts Wilf partitions, ranks A130091, conjugate A383512.
%Y A383532 A122111 represents conjugation in terms of Heinz numbers.
%Y A383532 A325324 counts integer partitions with distinct 0-appended differences, ranks A325367.
%Y A383532 A336866 counts non Wilf partitions, ranks A130092, conjugate A383513.
%Y A383532 A383709 counts Wilf partitions with distinct 0-appended differences, ranks A383712.
%Y A383532 Cf. A001223, A047966, A181819, A238745, A320348, A325325, A325349, A325366, A325368, A325388, A383506.
%K A383532 nonn
%O A383532 1,2
%A A383532 _Gus Wiseman_, May 15 2025