6 problems

For each base $b$, let $\pi_b(k)$ denote the number of base-$b$ lunar primes with $k$ digits. Applegate, Le Brun, and Sloane's conjecture. As $k$ tends to infinity, … This predicts…

Let $d_2(n)$ denote the number of binary lunar divisors of $n$, and let $111\ldots 111|_2$ be the binary number with $k$ digits all equal to $1$. A composition of $k$ is an ordered…

Let $d_2(n)$ denote the number of binary lunar divisors of $n$, and consider the odd $k$-digit numbers in base $2$. Le Brun et al.'s conjecture. If $k\geq 3$ and $k\neq 5$, the sec…

Let $d_2(n)$ denote the number of binary lunar divisors of $n$, and consider the odd $k$-digit numbers in base $2$. Le Brun et al.'s conjecture. Among all such odd numbers,…

Let $d_2(n)$ denote the number of binary lunar divisors of $n$, and let $111\ldots 10|_2$ denote the binary number with $k-1$ initial $1$'s followed by $0$. Le Brun et al.'s conjec…

Let $b\geq 3$ and let $n$ range over the $k$-digit numbers in base $b$. Write $d_b(n)$ for the number of lunar divisors of $n$ in base $b$, and write $111\ldots 1|_b$ for the base-…