We define the k-dimensional generalized Euler function \(\varphi _k(n)\) as the number of ordered k-tuples \((a_1,\ldots ,a_k)\in {\mathbb {N}}^k\) such that \(1\le a_1,\ldots ,a_k\le n\) and both the product \(a_1\cdots a_k\) and the sum \(a_1+\cdots +a_k\) are prime to n. We investigate some of the properties of the function \(\varphi _k(n)\), and obtain a corresponding Menon-type identity.

中文翻译：

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Euler算术函数和Menon恒等式的另一种推广

我们将k维广义Euler函数\（\ varphi _k（n）\）定义为{\ mathbb {N}} ^ k \中的有序k元组\（（a_1，\ ldots，a_k）\这样\（1 \ le a_1，\ ldots，a_k \ le n \）以及乘积\（a_1 \ cdots a_k \）和和\（a_1 + \ cdots + a_k \）都是n的素数。我们研究函数\（\ varphi _k（n）\）的某些属性，并获得相应的Menon型恒等式。