27 problems

Let $N=16n+6$ for some nonnegative integer $n$. Define $\a0\sigma_0(m)$ to be the number of positive divisors of $m$. The $N=16n+6$ congruence conjecture. … The authors report comp…

For a positive integer $n$, define … Let $p$ be the smallest prime factor of $n$. Lower-bound conjecture for $v(n)$. If … then … This would strengthen the proven bound…

Generalized perfect-number conjecture. One has

The coefficient characterization conjecture. Under this assumption, $A\equiv0\pmod4$ and $B\equiv2\pmod4$.

Let $a$ be a positive integer, let $n$ be a positive integer, and write $[q^k]F(q)$ for the coefficient of $q^k$ in $F(q)$. Let … be the sum of the divisors of $n$. Coefficient con…

Let $MO(t,n)$ denote the MacMahon-type arithmetic function used in the paper, and let $t,n$ be integers. MacMahon divisibility conjecture. If $t=10$, then … The paper presents this…

An integer $n$ is abundant if $\sigma(n)>2n$, and it is semiperfect if it equals the sum of some of its proper divisors. Odd-abundant semiperfectness conjecture. Every odd abundant…

Let $n$ be a practical number. A number is good $k$-abundant when it satisfies the paper's definition of that property, and a $k$-layered number is one admitting the corresponding…

For an integer $k\geq0$, let … denote the number of positive divisors of $n$. Merca's conjecture. If … for every $n\geq0$, then there is a sequence of prime numbers…

Let $k$ be a positive integer and $q$ a prime with $(k,q-1)=3$. Let $S_{k,q}(x)$ and $S'_{k,q}(x)$ denote the two sums considered in the paper, and compare their Landau and Ramanuj…

General weighted sine-sum oscillation conjecture.

Let … with the parameters and primed summation convention used in the paper. Cosine-sine oscillation conjecture. … This predicts unbounded oscillation at scale $x^{1/4}$. The motiv…

Let … and, for $m\in\mathbb{N}$, let $b_{m,n}$ denote the coefficient of $q^n$ in $f(q)^m$, with $b_{m,n}=0$ otherwise. Hong–Zhang's conjecture. There exists a constant $C>1$ such…

Connectedness conjecture. Given a superabundant number $n$, either there is a prime $p$ such that $np$ is superabundant or there is a prime $q$ such that $n/q$ is superabundant; mo…

Let $S$ be a 2-Mersenne prime and let $h\geq 2$. The polynomial sum-of-divisors function is denoted by $\sigma$. 2-Mersenne divisor conjecture. The integer $\sigma(S^{2h})$ is divi…

Let $M$ be a Mersenne prime and let $h\geq 2$. The polynomial sum-of-divisors function is denoted by $\sigma$. Mersenne divisor conjecture. The integer $\sigma(M^{2h})$ is divisibl…

For real $x$, let $N_e(x,n)$, $\bar{N}_e(x,n)$ and $\bar{\bar{N}}_e(x,n)$ count respectively the strong, weak and very weak even alpha numbers $n$ not exceeding $x$. Even alpha-num…

Let $\sigma(n)$ denote the sum-of-divisors function. For integers $k\geq 2$, $x\geq 3$, and $a\in\mathbb{Z}$ satisfying … consider the solutions $n\leq x$ of $\sigma(n)=kn+a$. Poly…

Arithmetic correlations conjecture. There are numbers $\phi<1$ and $\psi>0$ such that

Variance conjecture for divisor sums in arithmetic progressions. If $X,q\to\infty$ in such a way that $\frac{\log X}{\log q}\to c\in(0,k)$, then

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 $K$ be a number field, fix an ideal class representative $uinmathcal{I}K$, let $finmathcal{Z}K$, and let $\mathfrak{F}$ be a system of binary forms as in the paper. For an…

A positive integer $n$ is near-perfect if it is the sum of all its proper divisors except one proper divisor, called its redundant divisor. For a fixed integer $k\geq 1$, the redun…

Technical divisor-sum conjecture. Then

Heath-Brown's upper-bound conjecture. Uniformly for $( u,q)=1$ and $qtext{?}N^{2-\text{?}}$,