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A383513 Heinz numbers of non conjugate Wilf partitions.

%I A383513 #10 May 15 2025 08:23:23
%S A383513 6,12,18,21,24,30,36,42,48,54,60,63,65,66,70,72,78,84,90,96,102,105,
%T A383513 108,110,114,120,126,132,133,138,140,144,147,150,154,156,162,165,168,
%U A383513 174,180,186,189,192,198,204,210,216,220,222,228,231,234,238,240,246
%N A383513 Heinz numbers of non conjugate Wilf partitions.
%C A383513 First differs from A381433 in having 65.
%C A383513 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 A383513 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).
%H A383513 Gus Wiseman, <a href="/A325325/a325325.txt">Sequences counting and ranking integer partitions by the differences of their successive parts.</a>
%e A383513 The terms together with their prime indices begin:
%e A383513     6: {1,2}
%e A383513    12: {1,1,2}
%e A383513    18: {1,2,2}
%e A383513    21: {2,4}
%e A383513    24: {1,1,1,2}
%e A383513    30: {1,2,3}
%e A383513    36: {1,1,2,2}
%e A383513    42: {1,2,4}
%e A383513    48: {1,1,1,1,2}
%e A383513    54: {1,2,2,2}
%e A383513    60: {1,1,2,3}
%e A383513    63: {2,2,4}
%e A383513    65: {3,6}
%e A383513    66: {1,2,5}
%e A383513    70: {1,3,4}
%e A383513    72: {1,1,1,2,2}
%e A383513    78: {1,2,6}
%e A383513    84: {1,1,2,4}
%e A383513    90: {1,2,2,3}
%e A383513    96: {1,1,1,1,1,2}
%t A383513 prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A383513 Select[Range[100],!UnsameQ@@DeleteCases[Differences[Prepend[prix[#],0]],0]&]
%Y A383513 Partitions of this type are counted by A336866.
%Y A383513 The conjugate version is A130092, complement A130091.
%Y A383513 Including differences of 0 gives complement of A325367, counted by A325324.
%Y A383513 The strict case is the complement of A325388, counted by A320348.
%Y A383513 The complement is A383512, counted by A098859.
%Y A383513 Also forbidding distinct multiplicities gives A383531, counted by A383530.
%Y A383513 These are positions of non-strict rows in A383534, or nonsquarefree numbers in A383535.
%Y A383513 A000040 lists the primes, differences A001223.
%Y A383513 A048767 is the Look-and-Say transform, union A351294, complement A351295.
%Y A383513 A055396 gives least prime index, greatest A061395.
%Y A383513 A056239 adds up prime indices, row sums of A112798, counted by A001222.
%Y A383513 A122111 represents conjugation in terms of Heinz numbers.
%Y A383513 A239455 counts Look-and-Say partitions, complement A351293.
%Y A383513 A383507 counts partitions that are Wilf and conjugate Wilf, ranks A383532.
%Y A383513 A383709 counts Wilf partitions with distinct augmented differences, ranks A383712.
%Y A383513 Cf. A000720, A005117, A048768, A238745, A325325, A325351, A325355, A325366, A325368, A381431, A383506.
%K A383513 nonn
%O A383513 1,1
%A A383513 _Gus Wiseman_, May 13 2025