A383513 - OEIS

%I #10 May 15 2025 08:23:23

%S 6,12,18,21,24,30,36,42,48,54,60,63,65,66,70,72,78,84,90,96,102,105,

%T 108,110,114,120,126,132,133,138,140,144,147,150,154,156,162,165,168,

%U 174,180,186,189,192,198,204,210,216,220,222,228,231,234,238,240,246

%N Heinz numbers of non conjugate Wilf partitions.

%C First differs from A381433 in having 65.

%C 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 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 Gus Wiseman, <a href="/A325325/a325325.txt">Sequences counting and ranking integer partitions by the differences of their successive parts.</a>

%e The terms together with their prime indices begin:

%e     6: {1,2}

%e    12: {1,1,2}

%e    18: {1,2,2}

%e    21: {2,4}

%e    24: {1,1,1,2}

%e    30: {1,2,3}

%e    36: {1,1,2,2}

%e    42: {1,2,4}

%e    48: {1,1,1,1,2}

%e    54: {1,2,2,2}

%e    60: {1,1,2,3}

%e    63: {2,2,4}

%e    65: {3,6}

%e    66: {1,2,5}

%e    70: {1,3,4}

%e    72: {1,1,1,2,2}

%e    78: {1,2,6}

%e    84: {1,1,2,4}

%e    90: {1,2,2,3}

%e    96: {1,1,1,1,1,2}

%t prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];

%t Select[Range[100],!UnsameQ@@DeleteCases[Differences[Prepend[prix[#],0]],0]&]

%Y Partitions of this type are counted by A336866.

%Y The conjugate version is A130092, complement A130091.

%Y Including differences of 0 gives complement of A325367, counted by A325324.

%Y The strict case is the complement of A325388, counted by A320348.

%Y The complement is A383512, counted by A098859.

%Y Also forbidding distinct multiplicities gives A383531, counted by A383530.

%Y These are positions of non-strict rows in A383534, or nonsquarefree numbers in A383535.

%Y A000040 lists the primes, differences A001223.

%Y A048767 is the Look-and-Say transform, union A351294, complement A351295.

%Y A055396 gives least prime index, greatest A061395.

%Y A056239 adds up prime indices, row sums of A112798, counted by A001222.

%Y A122111 represents conjugation in terms of Heinz numbers.

%Y A239455 counts Look-and-Say partitions, complement A351293.

%Y A383507 counts partitions that are Wilf and conjugate Wilf, ranks A383532.

%Y A383709 counts Wilf partitions with distinct augmented differences, ranks A383712.

%Y Cf. A000720, A005117, A048768, A238745, A325325, A325351, A325355, A325366, A325368, A381431, A383506.

%K nonn

%O 1,1

%A _Gus Wiseman_, May 13 2025