Let $d_3(n)$ denote the 3-th divisor function. In this paper, we study the asymptotic formula of the sum

$$\sum _{\begin{array}{c}1⩽n_1,n_2⩽X^{\frac{1}{2}}\\ 1⩽n_3⩽X^{\frac{1}{k}}\end{array}}{d}_3(n_1^2+n_2^2+n_3^k)$$

with $n_1,n_2,n_3\in {\mathbb{Z}}^{+}$ and $k⩾3$ be an integer. Previously only the case of $k=2$ is studied.

In this paper, it is proved that every pair of large positive even integers satisfying some necessary conditions can be represented in the form of a pair of eight cubes of primes and 287 powers of $2$. This improves the previous result.

Let $d_k(n)$ denote the k-th divisor function. In this paper, we give the asymptotic formula of the sum

$$\sum _{\begin{array}{c}x-y\le n_i^r\le x+y\\ i=1,2,\dots ,l\end{array}}{d}_k(n_1^r+n_2^r+\dots +n_l^r),$$

where $n_1,n_2,\dots ,n_l\in {\mathbb{Z}}^{+}$, $k\ge 2$, $r\ge 3$ and $l>2^{r-1}$ are integers.