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A289997
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Numbers n whose trajectory under iteration of the map k -> (sigma(k)+phi(k))/2 never reaches a fraction (that is, either the trajectory reaches a prime, which is a fixed point, or diverges to infinity).
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7
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1, 2, 3, 5, 6, 7, 10, 11, 13, 17, 19, 21, 22, 23, 26, 27, 29, 30, 31, 37, 38, 39, 40, 41, 43, 44, 45, 46, 47, 51, 52, 53, 57, 58, 59, 60, 61, 65, 66, 67, 68, 71, 73, 74, 75, 79, 80, 82, 83, 89, 91, 92, 97, 101, 103, 106, 107, 109, 113, 114, 115, 116, 117, 126, 127, 131, 133, 134, 135, 136, 137
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OFFSET
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1,2
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COMMENTS
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Suggested by N. J. A. Sloane in a post "Iterating some number-theoretic functions" to the Seqfan mailing list.
The iteration arrives at a fixed point when k becomes a prime P, because sigma(P)=P+1 and phi(P)=P-1, hence k -> k.
It would be nice to have an independent characterization of these numbers (not involving the map in the definition). - N. J. A. Sloane, Sep 03 2017
Conjecturally, all terms of A291790 are in the sequence, because their trajectories (see example in A291789 for starting value 270) grow indefinitely. - Hugo Pfoertner, Sep 04 2017
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LINKS
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EXAMPLE
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126 is in the sequence, because the following iteration arrives at a fixed point:
k sigma(k) phi(k)
126 312 36 k->(sigma(k)+phi(k))/2, (312+36)/2=174
174 360 56 k->(sigma(k)+phi(k))/2, (360+56)/2=208
208 434 96
265 324 208
266 480 108
294 684 84
384 1020 128
574 1008 240
624 1736 192
964 1694 480
1087 1088 1086 k->(sigma(k)+phi(k))/2, (1088+1086)/2=1087
1087 1088 1086 ... loop
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CROSSREFS
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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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