3 problems

Let $a$, $b$, and $m$ be positive integers such that $\gcd(a,b,m)=1$, $a\ne b$, and $a+b\ne m$. Let $\mathrm{cp}_{a,b,m}(n)$ denote the number of $(a,b,m)$-copartitions of $n$. Hal…

Let $a$ and $m$ be positive integers with $m$ odd and $\gcd(a,m)=1$, and let $\mathrm{cp}_{a,a,m}(n)$ denote the number of $(a,a,m)$-copartitions of $n$. Half-density conjecture. T…

Let $a$ and $m$ be positive integers, with $m$ even and $a$ odd, and let $\mathrm{cp}_{a,a,m}(n)$ denote the number of $(a,a,m)$-copartitions of $n$. Parity-density conjecture. … T…