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A340605
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Heinz numbers of integer partitions of even positive rank.
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13
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5, 11, 14, 17, 21, 23, 26, 31, 35, 38, 39, 41, 44, 47, 49, 57, 58, 59, 65, 66, 67, 68, 73, 74, 83, 86, 87, 91, 92, 95, 97, 99, 102, 103, 104, 106, 109, 110, 111, 122, 124, 127, 129, 133, 137, 138, 142, 143, 145, 149, 152, 153, 154, 156, 157, 158, 159, 164, 165
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OFFSET
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1,1
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COMMENTS
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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.
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LINKS
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FORMULA
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EXAMPLE
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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)
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MATHEMATICA
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rk[n_]:=PrimePi[FactorInteger[n][[-1, 1]]]-PrimeOmega[n];
Select[Range[100], EvenQ[rk[#]]&&rk[#]>0&]
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CROSSREFS
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Note: Heinz numbers are given in parentheses below.
These partitions are counted by A101708.
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.
Cf. A006141, A024430, A056239, A112798, A340387, A340598, A340600, A340608, A340609, A340610, A340653.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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