Problem

Source: Serbia JBMO TST 2022 P3

Tags: number theory



Find all natural numbers $n$ for which the following $5$ conditions hold: $(1)$ $n$ is not divisible by any perfect square bigger than $1$. $(2)$ $n$ has exactly one prime divisor of the form $4k+3$, $k\in \mathbb{N}_0$. $(3)$ Denote by $S(n)$ the sum of digits of $n$ and $d(n)$ as the number of positive divisors of $n$. Then we have that $S(n)+2=d(n)$. $(4)$ $n+3$ is a perfect square. $(5)$ $n$ does not have a prime divisor which has $4$ or more digits.