Equal Sum and Product Problem III

Denote by $N(n)$ the number of integer solutions (x₁,\,x₂,… ,xₙ) of the equation x₁+x₂+…+xₙ=x₁x₂·…· xₙ such that x₁≥ x₂≥…≥ xₙ≥ 1, n ∈ Z⁺. The aim of this paper are is twofold: first we present an asymptotic formula for ∑2≤ n≤ xN(n), then we verify that the counting function $N(n)$ takes very large value compared to its average value.