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Another generalization of Euler’s arithmetic function and Menon’s identity
The Ramanujan Journal ( IF 0.6 ) Pub Date : 2021-02-16 , DOI: 10.1007/s11139-020-00353-z László Tóth
中文翻译:
Euler算术函数和Menon恒等式的另一种推广
更新日期:2021-02-17
The Ramanujan Journal ( IF 0.6 ) Pub Date : 2021-02-16 , DOI: 10.1007/s11139-020-00353-z László Tóth
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:
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。