10 problems

A positive integer is practical if every smaller positive integer can be written as a sum of distinct divisors of it. Let $P(x)$ denote the number of practical numbers up to $x$. M…

Conjecture on two polygonal summands. For $s\in\{4,5,6,7,8,10\}$, all natural numbers can be written as a sum of a practical number and two $s$-gonal numbers.

Conjecture on practical and polygonal sums. If $s>3$, $s \not\equiv 0\pmod{12}$, and $s \not\equiv 4\pmod{12}$, then all sufficiently large natural numbers can be written as a sum…

Let a practical number be a positive integer whose every smaller positive integer is a sum of distinct divisors of it. For each integer $n\ge 4$, consider the interval…

Let $a,b,c$ be positive integers with $2 mid ab$ and $2 divides c$. A practical number is a positive integer each of whose integers from $1$ through it can be written as a sum…

A practical number is a positive integer each of whose integers from $1$ through it can be written as a sum of distinct divisors. Sun's conjecture. There are infinitely many positi…

A practical number is a positive integer each of whose integers from $1$ through it can be written as a sum of distinct divisors. Sun's conjecture. Every odd integer greater than o…

A sandwich of the first kind is a triple $\{p-1,p,p+1\}$ in which $p$ is prime and both $p-1$ and $p+1$ are practical. Let $s_n$ be the central prime in the $n$th such sandwich. Fi…

Let $a_n$ denote the $n$th practical number, where a positive integer is practical if every integer from $1$ through it is a sum of distinct divisors of that integer. Practical-num…

Sparsity conjecture for additive endpoints. The values $\rho_m$ are usually zero: the set of natural numbers $m$ for which $\rho_m\ne 0$ has vanishing asymptotic density.