A340602 Heinz numbers of integer partitions of even rank.

1, 2, 5, 6, 8, 9, 11, 14, 17, 20, 21, 23, 24, 26, 30, 31, 32, 35, 36, 38, 39, 41, 44, 45, 47, 49, 50, 54, 56, 57, 58, 59, 65, 66, 67, 68, 73, 74, 75, 80, 81, 83, 84, 86, 87, 91, 92, 95, 96, 97, 99, 102, 103, 104, 106, 109, 110, 111, 120, 122, 124, 125, 126, 127

Offset: 1

Author

Gus Wiseman, Jan 21 2021

Keywords

• nonn

Comments

The Dyson rank of a nonempty partition is its maximum part minus its length. The rank of an empty partition is 0.

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.

Examples

			The sequence of partitions with their Heinz numbers begins:
     1: ()           31: (11)           58: (10,1)
     2: (1)          32: (1,1,1,1,1)    59: (17)
     5: (3)          35: (4,3)          65: (6,3)
     6: (2,1)        36: (2,2,1,1)      66: (5,2,1)
     8: (1,1,1)      38: (8,1)          67: (19)
     9: (2,2)        39: (6,2)          68: (7,1,1)
    11: (5)          41: (13)           73: (21)
    14: (4,1)        44: (5,1,1)        74: (12,1)
    17: (7)          45: (3,2,2)        75: (3,3,2)
    20: (3,1,1)      47: (15)           80: (3,1,1,1,1)
    21: (4,2)        49: (4,4)          81: (2,2,2,2)
    23: (9)          50: (3,3,1)        83: (23)
    24: (2,1,1,1)    54: (2,2,2,1)      84: (4,2,1,1)
    26: (6,1)        56: (4,1,1,1)      86: (14,1)
    30: (3,2,1)      57: (8,2)          87: (10,2)
		

Links

• FindStat, St000145: The Dyson rank of a partition

Crossrefs

Taking only length gives A001222.

Taking only maximum part gives A061395.

These partitions are counted by A340601.

The complement is A340603.

The case of positive rank is A340605.

- Rank -

A047993 counts partitions of rank 0 (A106529).

A101198 counts partitions of rank 1 (A325233).

A101707 counts partitions of odd positive rank (A340604).

A101708 counts partitions of even positive rank (A340605).

A257541 gives the rank of the partition with Heinz number n.

A324516 counts partitions with rank = maximum minus minimum part (A324515).

A340653 counts factorizations of rank 0.

A340692 counts partitions of odd rank (A340603).

- Even -

A024430 counts set partitions of even length.

A027187 counts partitions of even length (A028260).

A027187 (also) counts partitions of even maximum (A244990).

A034008 counts compositions of even length.

A035363 counts partitions into even parts (A066207).

A052841 counts ordered set partitions of even length.

A058696 counts partitions of even numbers (A300061).

A067661 counts strict partitions of even length (A030229).

A236913 counts even-length partitions of even numbers (A340784).

A339846 counts factorizations of even length.

Cf. A000041, A006141, A056239, A072233, A112798, A168659, A325134, A326836, A326845, A340386, A340387.

Programs

Mathematica

Select[Range[100],EvenQ[PrimePi[FactorInteger[#][[-1,1]]]-PrimeOmega[#]]&]

Formula

Either n = 1 or A061395(n) - A001222(n) is even.