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PDF(299 KB)
与Smarandache函数有关的一个方程的解的注记
A Note on Solutions of an Equation Relating to Smarandache Function
设φ（n）、S（n）分别表示正整数n的Euler函数和Smarandache函数，白海荣和廖群英在[Smarandache函数的几类相关方程的解，数学学报中文版， 2019， 62（2）：247-254]中称方程φ（n）=∑d|n S（d）只有两个解，分别为n=25和n=3×25.本文指出，这两个数均不是此方程的解，并指出其出错原因是因为他们对Möbius反转公式的错误理解所造成的.
For any positive integer n, let φ(n) be the Euler function and S(n) be the Smarandache function. Bai and Liao[On the solutions for several classes of equations related to the smarandache function, Acta Math. Sin., Chin. Series, 2019, 62(2):247-254] proved that the equation φ(n)= ∑d|n S(d) has only two solutions:n=25 and n=3×25. In this paper, we point out that both numbers are not solutions of the equation, and point out that this mistake was caused by that the authors misunderstood the Möbius inversion formula.
Euler函数 / Smarandache函数 / Mö / bius反转公式 {{custom_keyword}} /
Euler function / Smarandache function / Möbius inversion formula {{custom_keyword}} /
[1] Bai H. R., Liao Q. Y., On the solutions for several classes of equations related to the smarandache function, Acta Math. Sin., Chin. Series, 2019, 62(2):247-254.
[2] Liu Z., Shi P., On the solvability of the equation φ(n)= ∑d|n S(d) (in Chinese), Journal of Southwest University (Natural Science Edition), 2013, 35(6):54-58.
[3] Kenneth H. R., Elementary Number Theory and Its Applications, 4th Ed., Addison Wesley, Reading, MA, Longman, 2000.
安徽省高校自然科学重点项目（KJ2019A1）
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