Properties of terms of OEIS A342810

The OEIS sequence A342810 contains the numbers that divide the smallest number that has the sum of their digits. It is proved that if a term $x$ has the form $3^m \times y$ where $m \geq 2$, then all prime factors of $y$ are prime divisors of solutions to $10^n \equiv 1 \; ( \bmod n)$. It is also proved that if a term $x$ has the form $3^m \times p \times q$ where $m \geq 2$, where $p$ is a prime divisor of a solution to $10^n \equiv 1 \; ( \bmod n)$ and where $q$ is the product of all other factors of the prime factorisation of $x$, then all numbers $3^m \times p^i \times q$ are also terms of the sequence A342810 for any integer $i$.