5 problems

Let $d_{10}(n)$ denote the number of dismal divisors of $n$ in base $10$, and let $\sigma_{10}(n)$ denote the dismal sum of those divisors. Dismal divisor-sum inequality. For every…

Let $d_2(n)$ denote the number of dismal divisors of $n$, and let $n$ range over the odd $k$-digit binary numbers. Odd-binary divisor-count conjecture. The unique maximum of…

Let $d_2(n)$ denote the number of dismal divisors of the $k$-digit binary number $n$. Binary maximum-divisor conjecture. Among all $k$-digit numbers, the maximal value of $d_2(n)$…

Let $d_b(n)$ denote the number of dismal divisors of $n$ in base $b$, and let $n$ range over the $k$-digit base-$b$ numbers. Maximum-divisor conjecture. For every base $b\geq 3$,…

Let $\pi_b(k)$ denote the number of base $b$ dismal primes with $k$ digits. Prime-counting conjecture. As $k$ tends to infinity, … The conjecture predicts that asymptotically almos…