Why do even exponents occur more often if the residue is $5$ mod $7$?

Question

Here :

A conjecture about numbers of the form $10^{m}(2^{k}−1)+2^{k-1}−1$, where $m$ is the number of decimal digits of $ 2^{k-1}$.

the "ec-numbers" are introduced. The numbers emerge by concatenating $2^k-1$ and $2^{k-1}-1$. I generated the exponents in the range $[100\ 000,150\ 000]$ for which the numbers do not have a small prime factor and for which the residue modulo $7$ is $5$. The exponent $k$ was even in about $\frac{2}{3}$ of all those numbers.

Is there a reason based on the small factors of numbers of the above form with residue $5$ modulo $7$ ?

I looked at the conditions that a number of the above form is divisible by $11$ , by $13$ and so on, but I didn't find a reason.