

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
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OFFSET

1,2


COMMENTS

Suggested by N. J. A. Sloane in a post "Iterating some numbertheoretic 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)=P1, 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 numbertheoretic 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



