cp's OEIS Frontend

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).

%I A289997 #31 Dec 23 2024 14:53:45
%S A289997 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 A289997 44,45,46,47,51,52,53,57,58,59,60,61,65,66,67,68,71,73,74,75,79,80,82,
%U A289997 83,89,91,92,97,101,103,106,107,109,113,114,115,116,117,126,127,131,133,134,135,136,137
%N 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).
%C A289997 Suggested by N. J. A. Sloane in a post "Iterating some number-theoretic functions" to the Seqfan mailing list.
%C A289997 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 A289997 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 A289997 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 A289997 N. J. A. Sloane, <a href="https://web.archive.org/web/*/http://list.seqfan.eu/oldermail/seqfan/2017-September/017915.html">Iterating some number-theoretic functions</a>, Posting in Seqfan mailing list, Sep 3, 2017
%e A289997 126 is in the sequence, because the following iteration arrives at a fixed point:
%e A289997     k        sigma(k)     phi(k)
%e A289997    126         312          36  k->(sigma(k)+phi(k))/2, (312+36)/2=174
%e A289997    174         360          56  k->(sigma(k)+phi(k))/2, (360+56)/2=208
%e A289997    208         434          96
%e A289997    265         324         208
%e A289997    266         480         108
%e A289997    294         684          84
%e A289997    384        1020         128
%e A289997    574        1008         240
%e A289997    624        1736         192
%e A289997    964        1694         480
%e A289997   1087        1088        1086  k->(sigma(k)+phi(k))/2, (1088+1086)/2=1087
%e A289997   1087        1088        1086  ... loop
%Y A289997 Cf. A000203, A000010, A290001, A291789, A291790.
%Y A289997 Complement of A291791.
%K A289997 nonn
%O A289997 1,2
%A A289997 _Hugo Pfoertner_, Sep 03 2017