On Number of Ordered Pair of Positive Integers with Given Least Common Multiple
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Abstract
In this article, we present an expression for the number of ordered pairs of positive integer $m$ with prime factorization $p_1^{\alpha_1} \cdot p_2^{\alpha_3} \ldots p_n^{\alpha_n}$, and en route introduce an identity viz:
%\begin{equation*}
\begin{align*}
&\prod_{i=1}^n (2 \alpha_i+1) = \sum (\alpha_1+1) \alpha_2 \ldots \alpha_n + \sum (\alpha_1+1)(\alpha_2+1) \alpha_3 \ldots \alpha_n + \\
& \cdots + \sum (\alpha_1+1)(\alpha_2+1) \ldots (\alpha_r+1) \alpha_{r+1} \ldots \alpha_n + \cdots + \alpha_1 \alpha_2 \ldots \alpha_n
\end{align*}
%\end{equation*}
%\medskip
Moreover, we investigate the asymptotic behaviour of the mean and variance of the relative number of order pairs with some given least common multiple $m$. The notion of ordered pair is widely used in the fields of geometry, statistics, computing and programming languages.
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