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A383531 Heinz numbers of integer partitions that do not have distinct multiplicities (Wilf) or distinct nonzero 0-appended differences (conjugate Wilf).

%I A383531 #11 May 16 2025 18:50:51
%S A383531 6,21,30,36,42,60,65,66,70,78,84,90,102,105,110,114,120,126,132,133,
%T A383531 138,140,150,154,156,165,168,174,180,186,198,204,210,216,220,222,228,
%U A383531 231,234,238,240,246,252,258,264,270,273,276,280,282,286,294,300,306,308
%N A383531 Heinz numbers of integer partitions that do not have distinct multiplicities (Wilf) or distinct nonzero 0-appended differences (conjugate Wilf).
%C A383531 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 A383531 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 A383531 Equals A130092 /\ A383513.
%e A383531 The terms together with their prime indices begin:
%e A383531     6: {1,2}
%e A383531    21: {2,4}
%e A383531    30: {1,2,3}
%e A383531    36: {1,1,2,2}
%e A383531    42: {1,2,4}
%e A383531    60: {1,1,2,3}
%e A383531    65: {3,6}
%e A383531    66: {1,2,5}
%e A383531    70: {1,3,4}
%e A383531    78: {1,2,6}
%e A383531    84: {1,1,2,4}
%e A383531    90: {1,2,2,3}
%e A383531   102: {1,2,7}
%e A383531   105: {2,3,4}
%e A383531   110: {1,3,5}
%e A383531   114: {1,2,8}
%e A383531   120: {1,1,1,2,3}
%t A383531 prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n], {p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A383531 conj[y_]:=If[Length[y]==0,y, Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
%t A383531 Select[Range[100],!UnsameQ@@Length/@Split[prix[#]] && !UnsameQ@@Length/@Split[conj[prix[#]]]&]
%Y A383531 These partitions are counted by A383530.
%Y A383531 Negating both sides gives A383532, counted by A383507.
%Y A383531 A048767 is the Look-and-Say transform, union A351294, complement A351295.
%Y A383531 A055396 gives least prime index, greatest A061395.
%Y A383531 A056239 adds up prime indices, row sums of A112798, counted by A001222.
%Y A383531 A098859 counts Wilf partitions, ranks A130091, conjugate A383512.
%Y A383531 A122111 represents conjugation in terms of Heinz numbers.
%Y A383531 A325324 counts integer partitions with distinct 0-appended differences, ranks A325367.
%Y A383531 A336866 counts non Wilf partitions, ranks A130092, conjugate A383513.
%Y A383531 A383709 counts Wilf partitions with distinct 0-appended differences, ranks A383712.
%Y A383531 Cf. A001223, A047966, A181819, A238745, A320348, A325325, A325349, A325366, A325368, A325388, A381431, A383506.
%K A383531 nonn
%O A383531 1,1
%A A383531 _Gus Wiseman_, May 15 2025