[seqfan] Proof of conjecture needed.

[L. Edson Jeffery]

I have something like fifteen saved drafts related to the 3x+1 problem, all
of which have been sitting there since May. One of the sequences is A257480
(https://oeis.org/draft/A257480) which is rather nice I think. In that
draft I stated a conjecture (see Conjecture 1 below) which, as it turns
out, I cannot prove. I have been working on the proof for three months to
no avail. I need to get all of those sequences proposed for review, so I
hereby relinquish priority and invite anyone to try to prove the
conjecture. If anyone can prove it, then please post the proof here so
others can see it and we can compare the proofs. Please do not edit any of
my drafts yet.

The problem is as follows.

Let N denote the set of natural numbers and N_1 the set of odd natural
numbers. Let |y|_2 denote the 2-adic valuation of y.

Let the map F : N_1 -> N_1 be defined by

(1)  F(x) = (3*x + 1)/2^|3*x+1|_2.

Let F^(k)(x) denote the result of iterating F precisely k times and defined
by the recurrence

(2)  F^(k)(x) = F(F^(k-1)(x))   (k>0)

with initial condition F^(0)(x) = x. The function F is A075677 (
https://oeis.org/A075677).

I have a PDF I have been preparing for A257480. The first half of the paper
is devoted to rigorously (I hope) deriving the following function based
essentially on (1), (2) and a property of the array A257499 (
https://oeis.org/draft/A257499). (I omit the details here.)

Let the map S : N -> N be defined by

S(n) = (3 + (3/2)^|1+F(4*n-3)|_2*(1 + F(4*n-3))/6.

Let S^(k)(n) denote the result of iterating S precisely k times and defined
by the recurrence

(2)  S^(k)(n) = S(S^(k-1)(n))   (k>0)

with initial condition S^(0)(n) = n.

Conjecture 1. For all n > 1, log(S(n))/log(n) < log(3)/log(2).

The second half of the above-mentioned PDF is supposed to contain the proof
of Conjecture 1. For completeness, I state the following conjecture and
theorem.

Conjecture 2. For each n, there exists a k such that S^(k)(n) = 1.

Theorem 1. Conjecture 2 is equivalent to the 3x+1 conjecture.

Finally, I will gladly try to communicate the work I have done so far on
the proof to anyone who is interested.


Ed Jeffery