A340604 Heinz numbers of integer partitions of odd positive rank.

3, 7, 10, 13, 15, 19, 22, 25, 28, 29, 33, 34, 37, 42, 43, 46, 51, 52, 53, 55, 61, 62, 63, 69, 70, 71, 76, 77, 78, 79, 82, 85, 88, 89, 93, 94, 98, 101, 105, 107, 113, 114, 115, 116, 117, 118, 119, 121, 123, 130, 131, 132, 134, 136, 139, 141, 146, 147, 148, 151

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:
      3: (2)         46: (9,1)       82: (13,1)
      7: (4)         51: (7,2)       85: (7,3)
     10: (3,1)       52: (6,1,1)     88: (5,1,1,1)
     13: (6)         53: (16)        89: (24)
     15: (3,2)       55: (5,3)       93: (11,2)
     19: (8)         61: (18)        94: (15,1)
     22: (5,1)       62: (11,1)      98: (4,4,1)
     25: (3,3)       63: (4,2,2)    101: (26)
     28: (4,1,1)     69: (9,2)      105: (4,3,2)
     29: (10)        70: (4,3,1)    107: (28)
     33: (5,2)       71: (20)       113: (30)
     34: (7,1)       76: (8,1,1)    114: (8,2,1)
     37: (12)        77: (5,4)      115: (9,3)
     42: (4,2,1)     78: (6,2,1)    116: (10,1,1)
     43: (14)        79: (22)       117: (6,2,2)
		

Links

• FindStat, St000145: The Dyson rank of a partition

Crossrefs

Note: Heinz numbers are given in parentheses below.

These partitions are counted by A101707.

Allowing negative ranks gives A340692, counted by A340603.

The even version is A340605, counted by A101708.

The not necessarily odd case is A340787, counted by A064173.

A001222 gives number of prime indices.

A061395 gives maximum prime index.

- 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.

A340653 counts balanced factorizations.

- Odd -

A000009 counts partitions into odd parts (A066208).

A027193 counts partitions of odd length (A026424).

A027193 (also) counts partitions of odd maximum (A244991).

A058695 counts partitions of odd numbers (A300063).

A067659 counts strict partitions of odd length (A030059).

A160786 counts odd-length partitions of odd numbers (A300272).

A339890 counts factorizations of odd length.

A340101 counts factorizations into odd factors.

A340102 counts odd-length factorizations into odd factors.

A340385 counts partitions of odd length and maximum (A340386).

Cf. A001221, A006141, A056239, A112798, A168659, A200750, A316413, A325134, A340601, A340602, A340608, A340609, A340610.

Programs

Mathematica

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

Formula

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

A340604 \/ A340605 = A340787.