Let $p\colon \mathbb{Z} \rightarrow \mathbb{Z}$ be a polynomial whose leading coefficient is
positive and such that there exists no $d≥2$ with $d ∣ p(n)$ for all $n≥1$. Is it true that,
for all sufficiently large $m$, there exist integers $1≤n_1<\dots < n_k$ such that
$$1=\frac{1}{n_1}+\cdots+\frac{1}{n_k}$$
and
$$m=p(n_1)+\cdots+p(n_k)$$?
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