# Divisibility

## Distribution and upper bound of mimic numbers ★★

Author(s): Bhattacharyya

\begin{problem}

Let the notation $a|b$ denote ''$a$ divides $b$''. The mimic function in number theory is defined as follows [1].

\begin{definition} For any positive integer $\mathcal{N} = \sum_{i=0}^{n}\mathcal{X}_{i}\mathcal{M}^{i}$ divisible by $\mathcal{D}$, the mimic function, $f(\mathcal{D} | \mathcal{N})$, is given by,

$$f(\mathcal{D} | \mathcal{N}) = \sum_{i=0}^{n}\mathcal{X}_{i}(\mathcal{M}-\mathcal{D})^{i}$$

\end{definition}

By using this definition of mimic function, the mimic number of any non-prime integer is defined as follows [1].

\begin{definition} The number $m$ is defined to be the mimic number of any positive integer $\mathcal{N} = \sum_{i=0}^{n}\mathcal{X}_{i}\mathcal{M}^{i}$, with respect to $\mathcal{D}$, for the minimum value of which $f^{m}(\mathcal{D} | \mathcal{N}) = \mathcal{D}$. \end{definition}

Given these two definitions and a positive integer $\mathcal{D}$, find the distribution of mimic numbers of those numbers divisible by $\mathcal{D}$.

Again, find whether there is an upper bound of mimic numbers for a set of numbers divisible by any fixed positive integer $\mathcal{D}$. \end{problem}

Keywords: Divisibility; mimic function; mimic number