I'm curious to know if it is known some variants of the Erdős–Moser equation for different sequences of integers $a_m$ strictly increasing $1\leq a_1<a_2<\ldots<a_m<\ldots$
$$a_1^k+a_2^k+\ldots +a_m^k=a_{m+1}^k\tag{1}$$ being $k\geq 1$ an integer. I add here the Wikipedia Erdős–Moser equation.
Question 1. I would like to know if this problem $(1)$ is in the literature. If you know it from the litearature answer this as a reference request, and I try to search an read the statement of problems similar than $(1)$ for different incraasing sequences of integers $(a_k)_{k\geq 1}$ from the literature. Many thanks.
Other question is the following specialization of these variants of Erdős–Moser equation.
Question 2. Here again $m\geq 1$ and $k\geq 1$ are integers, does have any solution $(k,m)$ the equation $$s_1^k+s_2^k+\ldots +s_m^k=s_{m+1}^k\tag{2}$$ where $s_k$ denotes the sequence of semiprimes? Many thanks.
Here semiprime means a positive integer $s\geq 1$ that is that is the product of two prime numbers. See the Wikipedia Semiprime.