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A353316
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Heinz numbers of integer partitions that have a fixed point but whose conjugate does not (counted by A118199).
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3
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4, 8, 16, 27, 32, 45, 54, 63, 64, 81, 90, 99, 108, 117, 126, 128, 135, 153, 162, 171, 180, 189, 198, 207, 216, 234, 243, 252, 256, 261, 270, 279, 297, 306, 324, 333, 342, 351, 360, 369, 378, 387, 396, 405, 414, 423, 432, 459, 468, 477, 486, 504, 512, 513, 522
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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 together with their prime indices begin:
4: (1,1)
8: (1,1,1)
16: (1,1,1,1)
27: (2,2,2)
32: (1,1,1,1,1)
45: (3,2,2)
54: (2,2,2,1)
63: (4,2,2)
64: (1,1,1,1,1,1)
81: (2,2,2,2)
90: (3,2,2,1)
99: (5,2,2)
108: (2,2,2,1,1)
117: (6,2,2)
126: (4,2,2,1)
128: (1,1,1,1,1,1,1)
For example, the partition (3,2,2,1) with Heinz number 90 has a fixed point at the second position, but its conjugate (4,3,1) has no fixed points, so 90 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 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, A114088, A188674, A252464, A300788, A325163, A325169, A352831, A352828, A352829.
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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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