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A352827
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Heinz numbers of integer partitions y with a fixed point y(i) = i. Such a fixed point is unique if it exists.
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28
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2, 4, 8, 9, 15, 16, 18, 21, 27, 30, 32, 33, 36, 39, 42, 45, 51, 54, 57, 60, 63, 64, 66, 69, 72, 78, 81, 84, 87, 90, 93, 99, 102, 108, 111, 114, 117, 120, 123, 125, 126, 128, 129, 132, 135, 138, 141, 144, 153, 156, 159, 162, 168, 171, 174, 175, 177, 180, 183
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OFFSET
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1,1
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COMMENTS
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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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EXAMPLE
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The terms together with their prime indices begin:
2: (1)
4: (1,1)
8: (1,1,1)
9: (2,2)
15: (3,2)
16: (1,1,1,1)
18: (2,2,1)
21: (4,2)
27: (2,2,2)
30: (3,2,1)
32: (1,1,1,1,1)
33: (5,2)
36: (2,2,1,1)
39: (6,2)
42: (4,2,1)
45: (3,2,2)
51: (7,2)
54: (2,2,2,1)
For example, the partition (3,2,2) with Heinz number 45 has a fixed point at position 2, so 45 is in the sequence.
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MATHEMATICA
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pq[y_]:=Length[Select[Range[Length[y]], #==y[[#]]&]];
Select[Range[100], pq[Reverse[Flatten[Cases[FactorInteger[#], {p_, k_}:>Table[PrimePi[p], {k}]]]]]==1&]
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CROSSREFS
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* = unproved
A122111 represents partition conjugation using Heinz numbers.
A238352 counts reversed partitions by fixed points, rank statistic A352822.
A352828 counts strict partitions without a fixed point.
A352833 counts partitions by fixed points.
Cf. A062457, A064410, A065770, A093641, A257990, A342192, A352486, A352823, A352824 (characteristic function), A352825, A352831.
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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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