一类广义欧拉函数的准确计算公式

The Explicit Formula for a Special Class of Generalized Euler Functions

为将Lehmer同余式从模奇质数平方推广至模任意数的平方,Cai等(CAI T X, FU X D, ZHOU X. Acta Aritmetica,2002,103(3):203-214.)定义了广义欧拉函数φ e (n).最近Cai等给出了 e=3,4,6 时广义欧拉函数φ e (n)的计算公式.利用初等数论与组合的方法和技巧,完全确定了一类广义欧拉函数的计算公式,即给出当 e 为 n 的特殊正因数时,φ e (n)的准确计算公式,从而推广Cai等的相关主要结果,并由此给出φ e (n)为偶数的一个充分必要条件.

For a fixed positive integer n , in order to generalize the modulo from the square of prime numbers to the square of an arbitrary integer for the well known Lehmer congruence formula, Cai, et al(CAI T X, FU X D, ZHOU X. Acta Aritmetica,2002, 130(3):203-214.), defined the generalized Euler function φ e(n) in 2007 and then determined the explicit formulas for φ e(n)(e=3,4,6) in 2013 and 2016. The present paper continues the study, obtains the computing formula of φ e(n) for some special divisor e of n , which is a generalization for the corresponding results of Cai, et al, and then gives a sufficient and necessary condition for 2|φ e(n).