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A340604
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Heinz numbers of integer partitions of odd positive rank.
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23
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3, 7, 10, 13, 15, 19, 22, 25, 28, 29, 33, 34, 37, 42, 43, 46, 51, 52, 53, 55, 61, 62, 63, 69, 70, 71, 76, 77, 78, 79, 82, 85, 88, 89, 93, 94, 98, 101, 105, 107, 113, 114, 115, 116, 117, 118, 119, 121, 123, 130, 131, 132, 134, 136, 139, 141, 146, 147, 148, 151
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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:
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)
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MATHEMATICA
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rk[n_]:=PrimePi[FactorInteger[n][[-1, 1]]]-PrimeOmega[n];
Select[Range[100], OddQ[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 A101707.
A001222 gives number of prime indices.
- 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.
Cf. A001221, A006141, A056239, A112798, A168659, A200750, A316413, A325134, A340601, A340602, A340608, A340609, A340610.
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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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