OFFSET
1,1
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.
LINKS
EXAMPLE
The sequence of partitions with their Heinz numbers begins:
3: (2) 33: (5,2) 63: (4,2,2)
4: (1,1) 34: (7,1) 64: (1,1,1,1,1,1)
7: (4) 37: (12) 69: (9,2)
10: (3,1) 40: (3,1,1,1) 70: (4,3,1)
12: (2,1,1) 42: (4,2,1) 71: (20)
13: (6) 43: (14) 72: (2,2,1,1,1)
15: (3,2) 46: (9,1) 76: (8,1,1)
16: (1,1,1,1) 48: (2,1,1,1,1) 77: (5,4)
18: (2,2,1) 51: (7,2) 78: (6,2,1)
19: (8) 52: (6,1,1) 79: (22)
22: (5,1) 53: (16) 82: (13,1)
25: (3,3) 55: (5,3) 85: (7,3)
27: (2,2,2) 60: (3,2,1,1) 88: (5,1,1,1)
28: (4,1,1) 61: (18) 89: (24)
29: (10) 62: (11,1) 90: (3,2,2,1)
MATHEMATICA
Select[Range[100], OddQ[PrimePi[FactorInteger[#][[-1, 1]]]-PrimeOmega[#]]&]
CROSSREFS
Note: Heinz numbers are given in parentheses below.
These partitions are counted by A340692.
The case of positive rank is A340604.
- Rank -
A001222 gives number of prime indices.
A061395 gives maximum prime index.
A257541 gives the rank of the partition with Heinz number n.
A340653 counts balanced factorizations.
- Odd -
A339890 counts factorizations of odd length.
A340102 counts odd-length factorizations into odd factors.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jan 21 2021
STATUS
approved