论文标题
关于除数总和在算术进程中的功率值
On Power Values of Sum of Divisors function in Arithmetic Progressions
论文作者
论文摘要
令$ a \ geq 1，b \ geq 0 $和$ k \ geq 2 $为任何给定的整数。事实证明，存在无限的自然数$ m $，因此$ m $的除数总和是完美的$ k $ th功率。当$ m $的值属于任何给定的无限算术进程$ an+b $时，我们会尝试概括此结果。我们证明，如果$ a $相对为$ b $，并且$ b $ modulo $ a $的订单相对为$ k $，那么存在无限的许多自然数$ n $，因此$ an+b $的除数的总和是完美的$ k $ th功率。我们还证明，通常，对于任何天然数字$ n $而言，$ an+b $的除数总和都不是完美的$ k $ th功率，或者$ an+b $的除数总和是无限的许多自然数量$ n $的完美$ k $ th。
Let $a\geq 1, b\geq 0$ and $k\geq 2$ be any given integers. It has been proven that there exist infinitely many natural numbers $m$ such that sum of divisors of $m$ is a perfect $k$th power. We try to generalize this result when the values of $m$ belong to any given infinite arithmetic progression $an+b$. We prove if $a$ is relatively prime to $b$ and order of $b$ modulo $a$ is relatively prime to $k$ then there exist infinitely many natural numbers $n$ such that sum of divisors of $an+b$ is a perfect $k$th power. We also prove that, in general, either sum of divisors of $an+b$ is not a perfect $k$th power for any natural number $n$ or sum of divisors of $an+b$ is a perfect $k$th power for infinitely many natural numbers $n$.
