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 A289997 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). 7

%I

%S 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,

%T 44,45,46,47,51,52,53,57,58,59,60,61,65,66,67,68,71,73,74,75,79,80,82,

%U 83,89,91,92,97,101,103,106,107,109,113,114,115,116,117,126,127,131,133,134,135,136,137

%N 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).

%C Suggested by N. J. A. Sloane in a post "Iterating some number-theoretic functions" to the Seqfan mailing list.

%C 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.

%C 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

%C 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

%H N. J. A. Sloane, <a href="http://list.seqfan.eu/pipermail/seqfan/2017-September/017915.html">Iterating some number-theoretic functions</a>, Posting in Seqfan mailing list, Sep 3, 2017

%e 126 is in the sequence, because the following iteration arrives at a fixed point:

%e k sigma(k) phi(k)

%e 126 312 36 k->(sigma(k)+phi(k))/2, (312+36)/2=174

%e 174 360 56 k->(sigma(k)+phi(k))/2, (360+56)/2=208

%e 208 434 96

%e 265 324 208

%e 266 480 108

%e 294 684 84

%e 384 1020 128

%e 574 1008 240

%e 624 1736 192

%e 964 1694 480

%e 1087 1088 1086 k->(sigma(k)+phi(k))/2, (1088+1086)/2=1087

%e 1087 1088 1086 ... loop

%Y Cf. A000203, A000010, A290001, A291789, A291790.

%Y Complement of A291791.

%K nonn

%O 1,2

%A _Hugo Pfoertner_, Sep 03 2017

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Last modified October 17 08:36 EDT 2019. Contains 328107 sequences. (Running on oeis4.)