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A383512 Heinz numbers of conjugate Wilf partitions.

%I A383512 #10 May 15 2025 08:23:15
%S A383512 1,2,3,4,5,7,8,9,10,11,13,14,15,16,17,19,20,22,23,25,26,27,28,29,31,
%T A383512 32,33,34,35,37,38,39,40,41,43,44,45,46,47,49,50,51,52,53,55,56,57,58,
%U A383512 59,61,62,64,67,68,69,71,73,74,75,76,77,79,80,81,82,83,85
%N A383512 Heinz numbers of conjugate Wilf partitions.
%C A383512 First differs from A364347 in having 130 and lacking 110.
%C A383512 First differs from A381432 in lacking 65 and 133.
%C A383512 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 A383512 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 A383512 Gus Wiseman, <a href="/A325325/a325325.txt">Sequences counting and ranking integer partitions by the differences of their successive parts.</a>
%e A383512 The terms together with their prime indices begin:
%e A383512      1: {}           17: {7}            35: {3,4}
%e A383512      2: {1}          19: {8}            37: {12}
%e A383512      3: {2}          20: {1,1,3}        38: {1,8}
%e A383512      4: {1,1}        22: {1,5}          39: {2,6}
%e A383512      5: {3}          23: {9}            40: {1,1,1,3}
%e A383512      7: {4}          25: {3,3}          41: {13}
%e A383512      8: {1,1,1}      26: {1,6}          43: {14}
%e A383512      9: {2,2}        27: {2,2,2}        44: {1,1,5}
%e A383512     10: {1,3}        28: {1,1,4}        45: {2,2,3}
%e A383512     11: {5}          29: {10}           46: {1,9}
%e A383512     13: {6}          31: {11}           47: {15}
%e A383512     14: {1,4}        32: {1,1,1,1,1}    49: {4,4}
%e A383512     15: {2,3}        33: {2,5}          50: {1,3,3}
%e A383512     16: {1,1,1,1}    34: {1,7}          51: {2,7}
%t A383512 prix[n_]:=If[n==1,{}, Flatten[Cases[FactorInteger[n], {p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A383512 Select[Range[100], UnsameQ@@DeleteCases[Differences[Prepend[prix[#],0]],0]&]
%Y A383512 Partitions of this type are counted by A098859.
%Y A383512 The conjugate version is A130091, complement A130092.
%Y A383512 Including differences of 0 gives A325367, counted by A325324.
%Y A383512 The strict case is A325388, counted by A320348.
%Y A383512 The complement is A383513, counted by A336866.
%Y A383512 Also requiring distinct multiplicities gives A383532, counted by A383507.
%Y A383512 These are the positions of strict rows in A383534, or squarefree numbers in A383535.
%Y A383512 A000040 lists the primes, differences A001223.
%Y A383512 A048767 is the Look-and-Say transform, union A351294, complement A351295.
%Y A383512 A055396 gives least prime index, greatest A061395.
%Y A383512 A056239 adds up prime indices, row sums of A112798, counted by A001222.
%Y A383512 A122111 represents conjugation in terms of Heinz numbers.
%Y A383512 A239455 counts Look-and-Say partitions, complement A351293.
%Y A383512 A325349 counts partitions with distinct augmented differences, ranks A325366.
%Y A383512 A383530 counts partitions that are not Wilf or conjugate Wilf, ranks A383531.
%Y A383512 A383709 counts Wilf partitions with distinct augmented differences, ranks A383712.
%Y A383512 Cf. A000720, A005117, A048768, A109298, A325325, A325351, A325355, A325368, A381431, A383506.
%K A383512 nonn
%O A383512 1,2
%A A383512 _Gus Wiseman_, May 13 2025