[seqfan] Iterating some number-theoretic functions
Neil Sloane
njasloane at gmail.com
Sun Sep 3 06:49:28 CEST 2017
Dear Sequence Fans,
Let sigma = A203, phi = A10, psi = A1615. Richard Guy's Unsolved Problems
in Number Theory, 3rd ed., (UPNT) Section B41, pp. 147-150, mentions
several problems related to iterating these functions that caught my eye
recently.
(a) A039654 and many cross-referenced sequences deal with the trajectory of
n under the map k -> sigma(k)-1, and the open question of whether it always
reaches a prime. Some of these seqs need extending, or b-files.
Richard says Erdos studied this problem, and gives as reference the paper
Paul Erdos, Andrew Granville, Carl Pomerance and Claudia Spiro, <a
href="/A000010/a000010_1.pdf">On the normal behavior of the iterates of
some arithmetic functions</a>, Analytic number theory, BirkhĂ¤user Boston,
1990, pp. 165-204. [Annotated copy with A-numbers]
However, I can't seem to find any mention of the problem there - can
someone point me to the right page?
(b) If we iterate k -> (psi(k)+phi(k))/2 (Guy, p. 147) we get new sequences
A291784-A291787 which all need extending. I didn't add the sequence of n
such that the trajectory of n increases without limit (it begins 45, 50,
... but how does it continue?) Are there any starting values that go into a
nontrivial cycle?
(c) If we iterate k -> (sigma(k)+phi(k))/2 (same ref.), sometimes we reach
a fraction, when we say the trajectory has fractured, and we quit.
Question: what are the starting values n whos trajectory doesn't fracture?
Also, Richard asks if there are starting values which increase indefinitely
without fracturing.
On page 149 of UPNT Richard says that Erdos studied (b) and (c), and again
refers to the above paper - but again I couldn't see any mention of them
(it is a very long and technical paper and I didn't read it carefully).
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