一个包含算术函数S(d)和SL(d)的方程

An equation involving the arithmetical functions S(d) and SL(d)

目的研究一类包含算术函数S(n)及SL(n)方程的可解性。方法利用初等及组合方法。结果证明了所给方程有无穷多个正整数解,并给出了该方程所有解数的渐近公式。结论该方程的所有解为1,所有无平方因子数以及2乘所有无平方因子数。

Aim To study the solvability of an equation involving the arithmetical functions S（d） and SL（d）. Methods Using the elementary and combinatorial method. Results It was proved that the equation has infinite positive integer solutions,and get an asymptotic formula for its all positive integer solutions. Conclusion This shows that the solutions of the equation are 1,all square-free neumbers and all 2 times square-free numbers.