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A383712 Heinz numbers of integer partitions with distinct multiplicities (Wilf) and distinct 0-appended differences.

%I A383712 #7 May 16 2025 22:59:12
%S A383712 1,2,3,4,5,7,9,11,13,17,19,20,23,25,28,29,31,37,41,43,44,45,47,49,50,
%T A383712 52,53,59,61,67,68,71,73,75,76,79,83,89,92,97,98,99,101,103,107,109,
%U A383712 113,116,117,121,124,127,131,137,139,148,149,151,153,157,163,164
%N A383712 Heinz numbers of integer partitions with distinct multiplicities (Wilf) and distinct 0-appended differences.
%C A383712 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 A383712 Integer partitions with distinct multiplicities are called Wilf partitions.
%H A383712 Gus Wiseman, <a href="/A325325/a325325.txt">Sequences counting and ranking integer partitions by the differences of their successive parts.</a>
%F A383712 Equals A130091 /\ A325367.
%e A383712 The terms together with their prime indices begin:
%e A383712     1: {}
%e A383712     2: {1}
%e A383712     3: {2}
%e A383712     4: {1,1}
%e A383712     5: {3}
%e A383712     7: {4}
%e A383712     9: {2,2}
%e A383712    11: {5}
%e A383712    13: {6}
%e A383712    17: {7}
%e A383712    19: {8}
%e A383712    20: {1,1,3}
%e A383712    23: {9}
%e A383712    25: {3,3}
%e A383712    28: {1,1,4}
%e A383712    29: {10}
%e A383712    31: {11}
%e A383712    37: {12}
%e A383712    41: {13}
%e A383712    43: {14}
%e A383712    44: {1,1,5}
%e A383712    45: {2,2,3}
%e A383712    47: {15}
%e A383712    49: {4,4}
%e A383712    50: {1,3,3}
%t A383712 prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n], {p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A383712 Select[Range[100],UnsameQ@@Length/@Split[prix[#]] && UnsameQ@@Differences[Append[Reverse[prix[#]],0]]&]
%Y A383712 For just distinct multiplicities we have A130091 (conjugate A383512), counted by A098859.
%Y A383712 For just distinct 0-appended differences we have A325367, counted by A325324.
%Y A383712 These partitions are counted by A383709.
%Y A383712 A000040 lists the primes, differences A001223.
%Y A383712 A048767 is the Look-and-Say transform, union A351294, complement A351295.
%Y A383712 A055396 gives least prime index, greatest A061395.
%Y A383712 A056239 adds up prime indices, row sums of A112798, counted by A001222.
%Y A383712 A122111 represents conjugation in terms of Heinz numbers.
%Y A383712 A239455 counts Look-and-Say partitions, complement A351293.
%Y A383712 A336866 counts non Wilf partitions, ranks A130092, conjugate A383513.
%Y A383712 A383507 counts partitions that are Wilf and conjugate Wilf, ranks A383532.
%Y A383712 A383530 counts partitions that are not Wilf or conjugate-Wilf, ranks A383531.
%Y A383712 Cf. A000720, A005117, A047966, A238745, A320348, A325325, A325349, A325355, A325366, A325368, A325388, A383506.
%K A383712 nonn
%O A383712 1,2
%A A383712 _Gus Wiseman_, May 15 2025