47 problems

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…

Gabdullin–Konyagin–Iudelevich conjecture. There exists a constant $c_2$ with $0<c_2<\infty$ such that

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

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…

Let a positive integer $n$ be abundant if $sigma(n)>2n$, and call it weird if it is abundant but not $k$-near perfect for any positive integer $k$, where $n$ is $k$-near perfect wh…

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…

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…

Let $MO(a,t;n)$ denote the MacMahon-type sum of divisors considered in the paper. Amdeberhan–Andrews–Tauraso conjecture. For all $N\geq 0$, … This conjecture concerns a Ramanujan-l…

Fix an integer $m\geq2$, and consider 2-near perfect numbers of the form $2^kp^m$, where $k$ is an integer and $p$ is prime. Fixed-exponent finiteness conjecture. For any fixed…

Let a 2-near perfect number be a positive integer with the relevant divisor-sum property, and consider numbers of the form $2^kp^m$, where $k$ and $m$ are integers and $p$ is prime…

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…

Let $k$ be a positive integer, and suppose a $k$-layered number exists. Denote the smallest such number by $m_k$. Exponential growth conjecture. The size of $m_k$ grows exponential…

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…

Alaoglu–Erdős conjecture. For every superabundant number $n$, there exist primes $p,q$ such that both $np$ and $n/q$ are superabundant. Equivalently, the infinite lattice of supera…

Set $p=2$, and let $g(n,k,2)$ be defined by … For fixed $n$, let the maximum be taken over the relevant values of $k$. Asymptotic maximality conjecture. As $n$ increases, the maxim…

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 $@_{(1,1);2\mathbb{N}+1}$ denote the family of strong alpha numbers of order $(1,1)$ among the odd positive integers. Odd strong alpha-number conjecture. … The paper states tha…