Smarandache函数的几类相关方程的解

白海荣, 廖群英

数学学报 ›› 2019, Vol. 62 ›› Issue (2) : 247-254.

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PDF(425 KB)
数学学报 ›› 2019, Vol. 62 ›› Issue (2) : 247-254. DOI: 10.12386/A2019sxxb0022
论文

Smarandache函数的几类相关方程的解

    白海荣, 廖群英
作者信息 +

On the Solutions for Several Classes of Equations Related to the Smarandache Function

    Hai Rong BAI, Qun Ying LIAO
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摘要

φn),Sn)分别表示正整数n的Euler函数和Smarandache函数,利用初等的方法和技巧,依据Smarandache函数计算公式,给出k的方程φpαm)=Spαk)的所有解,其中p为素数,αm为正整数且gcd(m,p)=1,由此得到方程φn)=Snk)的所有解(n,k).进而确定了满足条件Sn)|σn)的全部正整数n.最后,根据莫比乌斯变换反演定理证明了方程φn)=Σd|n Sd)仅有两个解,分别为n=25n=3×25.

Abstract

Let φ(n),S(n) be the Euler function and the Smarandache function of the positive integer n, respectively. Based on elementary methods and techniques, according to the algorithm formula of the Smarandache function, all solutions of the equation φ(pαm)=S(pαk) are given, where p is a prime, α and m are both positive integers, and gcd(m,p)=1. And then we get the solutions of the equation φ(n)=S(nk). Furthermore, all positive integers n satisfying the condition S(n)|σ(n) are determined. At last, basing on the Mobius transformation inversion theorem, we prove that the equation φ(n)=Σd|n S(d) has only two solutions, namely, n=25 and n=3×25.

关键词

Smarandache函数 / 因数和函数 / Mobius变换

Key words

Smarandache function / function of divisor / Mobius transformation

引用本文

导出引用
白海荣, 廖群英. Smarandache函数的几类相关方程的解. 数学学报, 2019, 62(2): 247-254 https://doi.org/10.12386/A2019sxxb0022
Hai Rong BAI, Qun Ying LIAO. On the Solutions for Several Classes of Equations Related to the Smarandache Function. Acta Mathematica Sinica, Chinese Series, 2019, 62(2): 247-254 https://doi.org/10.12386/A2019sxxb0022

参考文献

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基金

国家自然科学基金资助项目(11401408);四川省科技厅资助项目(2016JY0134)

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