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) 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)
MATHEMATICA
rk[n_]:=PrimePi[FactorInteger[n][[-1, 1]]]-PrimeOmega[n];
Select[Range[100], OddQ[rk[#]]&&rk[#]>0&]
CROSSREFS
Note: Heinz numbers are given in parentheses below.
These partitions are counted by A101707.
A001222 gives number of prime indices.
A061395 gives maximum prime index.
- Rank -
A257541 gives the rank of the partition with Heinz number n.
A340653 counts balanced factorizations.
- Odd -
A339890 counts factorizations of odd length.
A340101 counts factorizations into odd factors.
A340102 counts odd-length factorizations into odd factors.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jan 21 2021
STATUS
approved