OFFSET
1,2
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.
LINKS
EXAMPLE
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)
MATHEMATICA
Select[Range[100], EvenQ[PrimePi[FactorInteger[#][[-1, 1]]]-PrimeOmega[#]]&]
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 -
A257541 gives the rank of the partition with Heinz number n.
A340653 counts factorizations of rank 0.
- Even -
A024430 counts set partitions of even length.
A034008 counts compositions of even length.
A052841 counts ordered set partitions of even length.
A339846 counts factorizations of even length.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jan 21 2021
STATUS
approved