A289997 - OEIS

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

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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