Let us call the numbers considered 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}$.
ec-numbers (named after Enzo Creti)
Hence if $d$ is the number of digits of the decimal expansion of $2^{n-1}-1$ , then $$ec(n)=(2^n-1)\cdot 10^d+2^{n-1}-1$$ for $n\ge 2$
How can we estimate the probability that for the numbers $n$ in the range $[a,b]$ , there is at least one prime of the form $ec(n)$ ? We can assume that $a$ and $b$ are "large".
I worked out the following : Apparently, $239$ out of $360$ ec-numbers do not have a prime factor $p\le 11$. $\frac{16}{77}$ of the positive integers do not have a prime factor $p\le 11$. Hence, the probability that $ec(n)$ is prime should be roughly $$\frac{239}{360}\cdot \frac{77}{16\cdot \ln(4^n)}\approx \frac{2.3}{n}$$ considering that for large $n$ we have $ec(n)\approx 4^n$
But the problem is that the events "$ec(m)$ is prime" and "$ec(n)$ is prime" are neither stochastically independent nor mutually exclusive.
