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A353317
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Heinz numbers of integer partitions that have a fixed point and a conjugate fixed point (counted by A188674).
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2
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2, 9, 15, 18, 21, 30, 33, 36, 39, 42, 51, 57, 60, 66, 69, 72, 78, 84, 87, 93, 102, 111, 114, 120, 123, 125, 129, 132, 138, 141, 144, 156, 159, 168, 174, 175, 177, 183, 186, 201, 204, 213, 219, 222, 228, 237, 240, 245, 246, 249, 250, 258, 264, 267, 275, 276
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OFFSET
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1,1
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COMMENTS
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A fixed point of a sequence y is an index y(i) = i. A fixed point of a partition is unique if it exists.
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 and their prime indices begin:
2: (1)
9: (2,2)
15: (3,2)
18: (2,2,1)
21: (4,2)
30: (3,2,1)
33: (5,2)
36: (2,2,1,1)
39: (6,2)
42: (4,2,1)
51: (7,2)
57: (8,2)
60: (3,2,1,1)
66: (5,2,1)
69: (9,2)
72: (2,2,1,1,1)
78: (6,2,1)
84: (4,2,1,1)
For example, the partition (2,2,1,1) with Heinz number 36 has a fixed point at the second position, as does its conjugate (4,2), so 36 is in the sequence.
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MATHEMATICA
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primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
pq[y_]:=Length[Select[Range[Length[y]], #==y[[#]]&]];
conj[y_]:=If[Length[y]==0, y, Table[Length[Select[y, #>=k&]], {k, 1, Max[y]}]];
Select[Range[100], pq[Reverse[primeMS[#]]]>0&& pq[conj[Reverse[primeMS[#]]]]>0&]
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CROSSREFS
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These partitions are counted by A188674.
Fixed point but no conjugate fixed point: A353316, counted by A118199.
A122111 represents partition conjugation using Heinz numbers.
A238352 counts reversed partitions by fixed points, rank statistic A352822.
A238394 counts reversed partitions without a fixed point, ranked by A352830.
A238395 counts reversed partitions with a fixed point, ranked by A352872.
A352826 ranks partitions w/o a fixed point, counted by A064428 (unproved).
A352827 ranks partitions with a fixed point, counted by A001522 (unproved).
Cf. A001222, A065770, A093641, A252464, A325039, A325163, A325169, A352828, A352831, A352832, A352833.
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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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