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:
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)
MATHEMATICA
rk[n_]:=PrimePi[FactorInteger[n][[-1, 1]]]-PrimeOmega[n];
Select[Range[100], EvenQ[rk[#]]&&rk[#]>0&]
CROSSREFS
Note: Heinz numbers are given in parentheses below.
These partitions are counted by A101708.
A001222 counts prime indices.
A061395 gives maximum prime index.
A072233 counts partitions by sum and length.
- Rank -
A257541 gives the rank of the partition with Heinz number n.
- Even -
A339846 counts factorizations of even length.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jan 21 2021
STATUS
approved