A383512 Heinz numbers of conjugate Wilf partitions.

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, 32, 33, 34, 35, 37, 38, 39, 40, 41, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 55, 56, 57, 58, 59, 61, 62, 64, 67, 68, 69, 71, 73, 74, 75, 76, 77, 79, 80, 81, 82, 83, 85

Offset

1,2

Comments

First differs from A364347 in having 130 and lacking 110.

First differs from A381432 in lacking 65 and 133.

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.

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).

Links

Table of n, a(n) for n=1..67.

Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

Example

The terms together with their prime indices begin:
     1: {}           17: {7}            35: {3,4}
     2: {1}          19: {8}            37: {12}
     3: {2}          20: {1,1,3}        38: {1,8}
     4: {1,1}        22: {1,5}          39: {2,6}
     5: {3}          23: {9}            40: {1,1,1,3}
     7: {4}          25: {3,3}          41: {13}
     8: {1,1,1}      26: {1,6}          43: {14}
     9: {2,2}        27: {2,2,2}        44: {1,1,5}
    10: {1,3}        28: {1,1,4}        45: {2,2,3}
    11: {5}          29: {10}           46: {1,9}
    13: {6}          31: {11}           47: {15}
    14: {1,4}        32: {1,1,1,1,1}    49: {4,4}
    15: {2,3}        33: {2,5}          50: {1,3,3}
    16: {1,1,1,1}    34: {1,7}          51: {2,7}

Mathematica

prix[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Select[Range[100], UnsameQ@@DeleteCases[Differences[Prepend[prix[#], 0]], 0]&]

Crossrefs

Partitions of this type are counted by A098859.

The conjugate version is A130091, complement A130092.

Including differences of 0 gives A325367, counted by A325324.

The strict case is A325388, counted by A320348.

The complement is A383513, counted by A336866.

Also requiring distinct multiplicities gives A383532, counted by A383507.

These are the positions of strict rows in A383534, or squarefree numbers in A383535.

A000040 lists the primes, differences A001223.

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

A055396 gives least prime index, greatest A061395.

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

A122111 represents conjugation in terms of Heinz numbers.

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

A325349 counts partitions with distinct augmented differences, ranks A325366.

A383530 counts partitions that are not Wilf or conjugate Wilf, ranks A383531.

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

Cf. A000720, A005117, A048768, A109298, A325325, A325351, A325355, A325368, A381431, A383506.

Sequence in context: A320340 A350838 A364347 * A228869 A088725 A094520

Adjacent sequences: A383509 A383510 A383511 * A383513 A383514 A383515

Keyword

nonn

Author

Gus Wiseman, May 13 2025

Status

approved