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A340931
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Heinz numbers of integer partitions of odd numbers into an odd number of parts.
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11
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2, 5, 8, 11, 17, 18, 20, 23, 31, 32, 41, 42, 44, 45, 47, 50, 59, 67, 68, 72, 73, 78, 80, 83, 92, 97, 98, 99, 103, 105, 109, 110, 114, 124, 125, 127, 128, 137, 149, 153, 157, 162, 164, 167, 168, 170, 174, 176, 179, 180, 182, 188, 191, 195, 197, 200, 207, 211
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OFFSET
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1,1
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COMMENTS
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The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This is 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 terms together with the corresponding partitions begins:
2: (1) 50: (3,3,1) 109: (29)
5: (3) 59: (17) 110: (5,3,1)
8: (1,1,1) 67: (19) 114: (8,2,1)
11: (5) 68: (7,1,1) 124: (11,1,1)
17: (7) 72: (2,2,1,1,1) 125: (3,3,3)
18: (2,2,1) 73: (21) 127: (31)
20: (3,1,1) 78: (6,2,1) 128: (1,1,1,1,1,1,1)
23: (9) 80: (3,1,1,1,1) 137: (33)
31: (11) 83: (23) 149: (35)
32: (1,1,1,1,1) 92: (9,1,1) 153: (7,2,2)
41: (13) 97: (25) 157: (37)
42: (4,2,1) 98: (4,4,1) 162: (2,2,2,2,1)
44: (5,1,1) 99: (5,2,2) 164: (13,1,1)
45: (3,2,2) 103: (27) 167: (39)
47: (15) 105: (4,3,2) 168: (4,2,1,1,1)
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MATHEMATICA
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primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Select[Range[100], OddQ[PrimeOmega[#]]&&OddQ[Total[primeMS[#]]]&]
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CROSSREFS
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Note: A-numbers of Heinz-number sequences are in parentheses below.
These partitions are counted by A160786.
The case of where the prime indices are also odd is A300272.
A072233 counts partitions by sum and length.
A112798 lists the prime indices of each positive integer.
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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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