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?

  • $\begingroup$ Comments are not for extended discussion; this conversation has been moved to chat. $\endgroup$
    – Xander Henderson
    Oct 13, 2022 at 16:10
  • $\begingroup$ Update Just to document a basic crosscheck: nothing between 10,000 and 1,000,000,000 (1.5 hrs, Delphi 64 bit) (previous crosscheck using numerical precision of 32bit has been insufficient on large numbers!) $\endgroup$ Oct 14, 2022 at 7:01

1 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 $a_1$.


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