Abstract

Given integers k ≥ 2 and ℓ ≥ 3, let S_{k,ℓ}^{∗} stand for the set of those positive integers n which can be written as n = p_{1}^{k}+p_{2}^{k}+…+p_{ℓ}^{k}, where p_{1}, p_{2}, …, p_{ℓ} are distinct prime factors of n. We study the properties of the sets S_{k,ℓ}^{∗} and we show in particular that, given any odd ℓ ≥ 3,

# ⋃_{k=2}^{∞} S_{k,ℓ}^{∗} = +∞.