Asymptotic Growth of A279688: Numbers n such n and 2n are anagrams in some base.

Question

I'm perplexed by the growth behavior of my recent OEIS Sequence A279688:

Numbers $n$ such $n$ and $2n$ are anagrams in some base.

Some examples of this sequence:

$a(2) = 8$ because in base 5, $8 = 13_5$ and $16 = 31_5$.

$a(3) = 18$ because in base $4$, $18 = 102_4$ and $36 = 210_4$.

I'm curious to hear:

1. A conjecture or proof of the asymptotic growth of A279688.
2. Any insight into the increases in slope that appear around ~2800 and ~9100.

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Here's a plot of the sequence for reference:

Answer 1

I think that I've gotten to the bottom of this!

Base-$3$ and base-$4$ are (perhaps unsurprisingly) the most common bases for $n$ and $2n$ to be anagrams.

I've published a new OEIS sequence which illustrates this well—A279916:

Least $b$ such that $A279688(n)$ and $2 \times A279688(n)$ are anagrams in base $b$.

If you look at the scatterplot of this new sequence, you can see that the steep slopes in the first graph coincide with places where no base-$3$ or base-$4$ anagrams appear (i.e. where anagrams are relatively rare.)

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Why do these gaps appear for base-3 and base-4?

One reason is that $n$ and $2n$ can only be anagrams in base-$b$ if there exists some integer $k$ where $b^k < n < \frac{1}{2}b^{k+1}$. If $n$ is outside of this range, then $n$ and $2n$ will have a different number of base-$b$ digits, and therefore cannot be anagrams of one another.

If you look at the original graph, the first steep slope appears where the base-3 and base-4 "forbidden zones" coincide—where the $y$-axis is in the interval:

$$\left(\frac{1}{2}4^8, 4^8\right) \cap \left(\frac{1}{2}3^{10}, 3^{10}\right) = (32768, 59049)$$