Let be an odd prime number and be a positive integer. Let , , and denote the -adic valuation of the integer , the set of positive integers, and the set of positive rational numbers, respectively. In this paper, we introduce an arithmetic function defined by for any positive integer . We show several interesting arithmetic properties about that function and then use them to establish some curious results involving the -adic valuation. Some of these results extend Farhi’s results from the case of even prime to that of odd prime.

中文翻译：

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好奇算术函数的进一步结果

设一个奇数质数，一个正整数。让，，并表示-整数的ADIC估值，在正整数的集合，并且组正有理数，分别。在本文中，我们介绍了一种算术函数 被定义为 对于任何正整数。我们显示有关该功能的一些有趣的算术属性，然后用它们来建立涉及到一些奇怪的结果- ADIC的估值。其中一些结果将Farhi的结果从偶数素数的情况扩展到了奇数素数的情况。