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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
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

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

LINKS

Table of n, a(n) for n=1..71.

N. J. A. Sloane, Iterating some number-theoretic functions, Posting in Seqfan mailing list, Sep 3, 2017

EXAMPLE

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

CROSSREFS

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

Complement of A291791.

Sequence in context: A085118 A276579 A166158 * A137313 A246867 A028805

Adjacent sequences:  A289994 A289995 A289996 * A289998 A289999 A290000

KEYWORD

nonn

AUTHOR

Hugo Pfoertner, Sep 03 2017

STATUS

approved

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Last modified August 22 20:34 EDT 2019. Contains 326186 sequences. (Running on oeis4.)