14 problems

Churchhouse's conjecture. For every positive integer $n$,

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…

Positive-density conjecture. The set $Q_1$ is infinite. Moreover, $Q_1$ has positive natural density in $\mathbb{N}$. This is suggested by computations showing many such representa…

Let $(fn){n einmathbb{N}{+}}$ be the increasing sequence whose range is $S'1$. A sequence is $m$-regular when its $m$-kernel generates a finitely generated module over the relevant…

Let $c_m(n)$ be the $m$-colored binary partition function and define, for $m\in\mathbb{N}_{+}$, … Boundedness conjecture. If $k\in\mathbb{N}_{+}$ and $m=2^{2k}-1$, then the sequenc…

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…

Equality conjecture. The sixth binary partition function is equal, aside from its initial zero term, to the partial sums of the Stern–Brocot sequence.

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…

Define a sequence $(a_n)_{n\geq 1}$ by $a_1=1$ and, for $n>1$, … where … Let $b_n$ be the number of binary partitions of $2n$ into powers of $2$, that is, the number of binary part…

Let $\mathcal A$ be a finite set of nonnegative integers, let $T$ be the period of its generating polynomial, and write the associated polynomial as … where $\mathcal A'$ is the fi…

Let $f_{\mathbb N,2}(n)$ denote the number of binary partitions of $n$, and let $\nu_2(m)$ denote the largest power of $2$ dividing $m$. Churchhouse's conjecture. For every even in…

Let $f_{\mathbb N,2}(n)$ denote the number of binary partitions of $n$, and let $\nu_2(m)$ denote the largest power of $2$ dividing $m$. Churchhouse's conjecture. For $n\geq 2$,…