5 problems

For binary partitions, let $\operatorname{num}_{\mathcal B}(n,x)$ denote the binary numerator. Ballantine–Beck–Feigon–Maurischat's binary unimodality conjecture. For every $n>5$, t…

For binary partitions, let $\operatorname{num}_{\mathcal B}(n,x)$ and $\operatorname{den}_{\mathcal B}(n,x)$ denote the corresponding numerator and denominator, where binary parts…

Let $\mathcal{T}(x)$ count the integers $n\leq x$ such that $b(n)$ is a sum of two squares, where $b(n)$ is the binary partition function. Logarithmic-density conjecture. There exi…

Let $c_m(n)$ denote the $m$-colored binary partition function, and let $u2$ be the 2-adic valuation. For positive integers $k,n$ and nonnegative integers $k$ as specified below, c…

Let $b_{2}(n)$ denote the number of binary partitions of $n$. Alkauskas's conjecture. For every $h\in\mathbb{N}$ with $8\nmid h$, there exist infinitely many $n$ such that $h$ divi…