Near misses to the start of a potential loop in the Collatz sequence

Question

A355239 contains numbers $k>4$ whose Collatz trajectory visits $k-1$ or $k+1$. These can be considered as near misses to the start of a second potential loop in the Collatz sequence. The last term is 9233. I've searched for further terms, but couldn't find any up to 31,100,000,000. Can anyone find any further terms?

Answer 1

The question of that type of cycle-near-misses ("cnm") is not much different from the general question of cycles in the Collatz-problem. We don't have a complete solution, but at least we have upper bounds for the members of a possible cycle, for which always a solution in rational numbers is present.

Let $N$ denote the number of odd steps ($3x+1$) and $S$ the number of even steps $x/2$, which is also the "$S$"um of the exponents at $2$ in the "Syracuse"-notation, which I usually prefer in my analyses.

1) Evaluation-scheme for general cycles

For a cycle the minimal element $a_1$ (we take the minimal element as leading one) we have the upper bound from the calculation of the rational solution, when all elements $a_1,a_2,...,a_N$ are equal to some mean-value $\alpha$ : $$ 2^S= (3+\frac1{\alpha})^N \qquad \implies \qquad \alpha = {1 \over 2^{S/N}-3} \qquad \implies \qquad a_1 \lt \alpha \tag 1$$ and so we do not need to search for arbitrary large elements $a_k$ to check whether a cycle of a certain length $N$ is possible.

Now the approximation of $2^{S/N}$ to $3$ is much jittering with increasing $N$ and might go near to zero so the "mean"-value $\alpha$ might go to very large values. However, there are bounds for the nearest approximation of $2^S-3^N$ or in our formulation $2^{S/N}-3$ known, for instance given by J. Ellison or G. Rhin (which we frequently refer to here in MSE). That bounds are however weak: they allow very large $\alpha$ . I use often a conjectured sharper upper bound which I verified for $N$ up to a million digits given by $$ { 1\over c \cdot N \cdot \ln N} \lt S \ln2 - N \ln 3 \tag 2 $$ For extremely large $N$ there are peaks which need $c=10$ for the rhs to fall above the lower bounds, but for $N \lt 10^{74} $ (I've done a quick check using the first $160$ convergents of the cont.frac. of $\log_2(3)$) we need only $c=2.5$ .
From this estimate we can conclude an upper bound for $\alpha(N)$ as $$ \alpha(N) \le c \cdot N^2 \ln N \tag {2a} $$ which means, for cycles of length $N=100$ odd steps we need only check $a_1 \lt 116 \; 000$ say, for $N=1000$ we get $a_1 \lt 17\; 300\; 000$ and so on.

2) Evaluation for "cycle-near-misses"

Now your modification towards "cycle-near-misses" gives a small modification in the equation (1). We need to include the $a_1+1$ or $a_1-1$ as the near-miss of $a_1$ arriving at the following form for some new mean value $\alpha_1$: $$ 2^S= (1+\frac1{\alpha_1})\cdot (3+\frac1{\alpha_1})^N \tag 3$$ (the derivation is not difficult).

Unfortunately I couldn't do a nice closed form for the calculation, so I did numerical search. Noticing that $\alpha_1 > \alpha$, $\lim_{N \to \infty} {\alpha_1(N) \over \alpha(N)} =1$ I found that $$\alpha(N) \lt \alpha_1(N) \lt 1.6 \alpha(N) \tag 4$$

From this, the upper bound for any smallest element in a cnm can be given in terms of the length $N$ of an assumed cnm as $$a_1 \lt 1.6 \cdot 2.5 \cdot N^2 \cdot \ln(N) \tag 5$$ for $N \lt 10^{74}$ and having only exceptional cases needing $c=10$ for $N \lt 10^{10^6}$

3) Conclusion using your given data

This small calculation does not give a disproof of the existence of cycle-near-misses, but after you've tested elements $a_1 \lt 31.1 \cdot 10^9$ we get -with all that rough estimates- that no cnm with length $N \lt 27572$ can exist except that which you have found in small numbers $N$ and $a_1$.