A340605 Heinz numbers of integer partitions of even positive rank.

5, 11, 14, 17, 21, 23, 26, 31, 35, 38, 39, 41, 44, 47, 49, 57, 58, 59, 65, 66, 67, 68, 73, 74, 83, 86, 87, 91, 92, 95, 97, 99, 102, 103, 104, 106, 109, 110, 111, 122, 124, 127, 129, 133, 137, 138, 142, 143, 145, 149, 152, 153, 154, 156, 157, 158, 159, 164, 165

Offset: 1

Author

Gus Wiseman, Jan 21 2021

Keywords

• nonn

Comments

The Dyson rank of a nonempty partition is its maximum part minus its number of parts. 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:
      5: (3)         57: (8,2)       97: (25)
     11: (5)         58: (10,1)      99: (5,2,2)
     14: (4,1)       59: (17)       102: (7,2,1)
     17: (7)         65: (6,3)      103: (27)
     21: (4,2)       66: (5,2,1)    104: (6,1,1,1)
     23: (9)         67: (19)       106: (16,1)
     26: (6,1)       68: (7,1,1)    109: (29)
     31: (11)        73: (21)       110: (5,3,1)
     35: (4,3)       74: (12,1)     111: (12,2)
     38: (8,1)       83: (23)       122: (18,1)
     39: (6,2)       86: (14,1)     124: (11,1,1)
     41: (13)        87: (10,2)     127: (31)
     44: (5,1,1)     91: (6,4)      129: (14,2)
     47: (15)        92: (9,1,1)    133: (8,4)
     49: (4,4)       95: (8,3)      137: (33)
		

Links

• FindStat, St000145: The Dyson rank of a partition

Crossrefs

Note: Heinz numbers are given in parentheses below.

Allowing any positive rank gives A064173 (A340787).

The odd version is counted by A101707 (A340604).

These partitions are counted by A101708.

The not necessarily positive case is counted by A340601 (A340602).

A001222 counts prime indices.

A061395 gives maximum prime index.

A072233 counts partitions by sum and length.

- Rank -

A047993 counts partitions of rank 0 (A106529).

A064173 counts partitions of negative rank (A340788).

A064174 counts partitions of nonnegative rank (A324562).

A064174 (also) counts partitions of nonpositive rank (A324521).

A101198 counts partitions of rank 1 (A325233).

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

A340692 counts partitions of odd rank (A340603).

- Even -

A027187 counts partitions of even length (A028260).

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

A035363 counts partitions into even parts (A066207).

A058696 counts partitions of even numbers (A300061).

A067661 counts strict partitions of even length (A030229).

A339846 counts factorizations of even length.

Cf. A006141, A024430, A056239, A112798, A340387, A340598, A340600, A340608, A340609, A340610, A340653.

Programs

Mathematica

rk[n_]:=PrimePi[FactorInteger[n][[-1,1]]]-PrimeOmega[n];
Select[Range[100],EvenQ[rk[#]]&&rk[#]>0&]

Formula

A061395(a(n)) - A001222(a(n)) is even and positive.

A340604 \/ A340605 = A340787.