  In this paper, we study a class of Cantor-integers $\{C_n\}_{n\geq 1}$ with
the base conversion function $f:\{0,\dots,m\}\to \{0,\dots,p\}$ being strictly
increasing and satisfying $f(0)=0$ and $f(m)=p$. Firstly we provide an
algorithm to compute the superior and inferior of the sequence
$\left\{\frac{C_n}{n^{\alpha}}\right\}_{n\geq 1}$ where $\alpha
=\log_{m+1}^{p+1}$, and obtain the exact values of the superior and inferior
when $f$ is a class of quadratic function. Secondly we show that the sequence
$\left\{\frac{C_n}{n^{\alpha}}\right\}_{n\geq 1}$ is dense in the close
interval with the endpoints being its inferior and superior respectively. As a
consequence, (i) we get the upper and lower pointwise density
$1/\alpha$-density of the self-similar measure supported on $\mathfrak{C}$ at
$0$, where $\mathfrak{C}$ is the Cantor set induced by Cantor-integers. (ii)
the sequence $\{\frac{C_n}{n^{\alpha}}\}_{n\geq 1}$ does not have cumulative
distribution function but have logarithmic distribution functions (given by a
specific Lebesgue integral). Lastly we obtain the Mellin-Perron formula for the
summation function of Cantor-integers. In addition, we investigate some
analytic properties of the limit function induced by Cantor-integers.
