30 problems

Let $\varphi(n)$, $\psi(n)$ and $\sigma(n)$ denote the Euler totient, Dedekind psi and sum-of-divisors functions, respectively. For integers $k\geq1$ and $n\geq2$, form the three q…

Let $\varphi(n)$, $\psi(n)$ and $\sigma(n)$ denote the Euler totient, Dedekind psi and sum-of-divisors functions, respectively. For integers $k\geq1$ and $n\geq2$, use…

Auxiliary solution-count conjectures.

Let $g_\ell(n)$ denote the number of subgroups of index $n$ in $\mathbb{Z}^\ell$, and write … Call a sequence $(a_k)_{k\in\mathbb{N}}$ log-concave at $k\geq 1$ when…

McCranie's conjecture. Every composite integer $n$ satisfying

Let $k$ be a positive integer, let $n$ range over positive integers, and let $\sigma(n)$ denote the sum of the positive divisors of $n$. Generalized Erdős–Sierpiński conjecture. Fo…

The lucky divisibility functions are defined by … where $\overset{*}{\omega}(n)$ counts distinct lucky divisors and $\overset{*}{\Omega}(n)$ counts them with their lucky order. Luc…

Asymptotic formula.

Bounded-height asymptotic conjecture. Then

Let $x,y\in\mathbb{N}$ be coprime and satisfy $x\ne y$. Distinctness conjecture. The values … are distinct. Here $\sigma$ is the sum-of-divisors function. Distinctness is known for…

Let $\sigma(x)$ denote the sum-of-divisors function for $x\in\mathbb{N}$. Distinctness conjecture. All values of … for $x\in\mathbb{N}$ are pairwise distinct. No equal values are k…

Eventual vanishing conjecture for $M(a,b)$. For each fixed $a>0$, there is a sufficiently large $b$ such that

For $b>0$, let $S_{b,m}$ denote the set of inputs associated with the nim-value $f(a,b,m)$ in SALIQUANT. The density conjecture for $S_{b,m}$. If $m=0$, then $S_{b,m}$ has positive…

EGPS conjecture. The preimage $s^{-1}(\mathcal{A})$ also has asymptotic density zero.

For $k$ with $0<k<n$, define $A_{2k}=\{n\in\mathbb{N}:\nu(n)=2k\}$, where $\nu(n)$ is the number of positive divisors of $n$. The divisor-class polynomial conjecture. No polynomial…

Consider the divide-and-residue game, and let ${\mathcal{SG}}(n)$ denote the nim-value of a heap of size $n$. Divide-and-residue conjecture. Every nim-value occurs for at least one…

For a positive integer $n=\prod_{i=1}^k p_i^{a_i}$, define … A Ruth–Aaron triple is a sequence $n,n+1,n+2$ of consecutive integers with equal values of $f$. Finite Ruth–Aaron tripl…

For a positive integer $n=\prod_{i=1}^k p_i^{a_i}$, define … A Ruth–Aaron number is an integer $n$ satisfying $f(n)=f(n+1)$. Erdős's infinitude conjecture. There are infinitely man…

Let … S{varphi}(x):=sum{nleq x}varphileft(leftlfloor x/n rightrfloorright). … S{varphi}(x)=dfrac{xlog(x)}{zeta(2)}(1+o(1)). … denotes the Riemann zeta function. The conjecture pred…

Let $p(n)$ denote the partition function, let $\mu(n)$ denote the Möbius function, and let $s_{n,1}^{(-1)}$ denote the entries in the first column of the inverse matrix under consi…

Let $n$ be an integer satisfying $n\le 2$. Existence conjecture for additive uniqueness sets. There exist $E\subseteq\mathbb{N}$ and a set $S$ of arithmetic functions such that $E$…

Let $E$ be a set and $S$ a set of arithmetic functions. For positive integers $k$ and $ell$, let $n(E,S,k)$ denote the number of functions in $S$ determined by the relation … for a…

Anavi–Pollack–Pomerance conjecture. Uniformly for integers $k$ satisfying $|k|\leq x/2$, the number of sporadic solutions $n\leq x$ is at most $(\log x)^{O(1)}$.

Let $k\in\mathbb{N}$ and let $f:\mathbb{N}\to\mathbb{C}$ satisfy … for all $u,v\in\mathbb{N}$. Characterization conjecture. Exactly one of the following possibilities holds: (1)…

Let $R_2$ be the function introduced in the paper, and let $x$ be an odd integer greater than $3$. The logarithmic lower-bound conjecture. … The conjecture proposes a lower bound f…