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- W3008346375 abstract "Let $a, b$ and $n$ be positive integers and $S = left{ {x_1, ..., x_n} right}$ be a set of $n$ distinct positive integers. The set $S$ is called a divisor chain if there is a permutation $sigma $ of ${1, ..., n}$ such that $x_{sigma (1)}|...|x_{sigma (n)}$. We say that the set $S$ consists of two coprime divisor chains if we can partition $S$ as $S = S_1cup S_2$, where $S_1$ and $S_2$ are divisor chains and each element of $S_1$ is coprime to each element of $S_2$. For any arithmetic function $f$, we define the function $f^a$ for any positive integer $x$ by $f^a(x): = (f(x))^a$. The matrix $(f^a(S))$ is the $ntimes n$ matrix having $f^a$ evaluated at the the greatest common divisor of $x_{i}$ and $x_{j}$ as its $(i, j)$-entry and the matrix $(f^a[S])$ is the $ntimes n$ matrix having $f^a$ evaluated at the least common multiple of $x_i$ and $x_j$ as its $(i, j)$-entry. In this paper, when $f$ is an integer-valued arithmetic function and $S$ consists of two coprime divisor chains with $1 notin S$, we establish the divisibility theorems between the determinants of the power matrices $(f^a(S))$ and $(f^b(S))$, between the determinants of the power matrices $(f^a[S])$ and $(f^b[S])$ and between the determinants of the power matrices $(f^a(S))$ and $(f^b[S])$. Our results extend Hong's theorem obtained in 2003 and the theorem of Tan, Lin and Liu gotten in 2011." @default.
- W3008346375 created "2020-03-06" @default.
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- W3008346375 date "2020-01-01" @default.
- W3008346375 modified "2023-09-27" @default.
- W3008346375 title "Divisibility among determinants of power matrices associated with integer-valued arithmetic functions" @default.
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- W3008346375 doi "https://doi.org/10.3934/math.2020130" @default.
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