A subset $S\subset\mathbb{N}$ contains the numbers $5$ and $8$.
If $n\in S$, then $7n-3$ and $7n+5$ are also in $S$
If $n\in S$, and $n=5m+4$, then $11m+9$ is also in $S$.
Numerically, $|S\cap[1,N]|=O(N^{0.43})$ The power law is plotted on the same graph as the count of S.
What is the correct value of the exponent?
This comes from For which $n\in\Bbb N$ can we divide $\{1,2,3,...,3n\}$ into $n$ subsets each with $3$ elements such that in each subset $\{x,y,z\}$ we have $x+y=3z$?, where larger solutions can be constructed from smaller ones
To close the question and take it off the unanswered list
Using Wolfie, the solution to $$\frac2{7^x}+\frac{1/5}{2.2^x}=1$$ is $$x\approx 0.434878$$