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).

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

Offset: 1

Author

Hugo Pfoertner, Sep 03 2017

Keywords

• nonn

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

Examples

			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
		

Links

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

Crossrefs

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

Complement of A291791.