包含Smarandache对偶函数的方程的正整数解

Positive integer solutions of an equation involving Smarandache dual function

利用初等数论及组合方法研究了一个包含Smarandache对偶函数及素因子函数方程∑d|n1/S*(d)=2Ω(n)的可解性.给出了这个方程所有正整数解的具体形式,即证明了该方程所有偶数解为n=24.330、n=26.312、n=8p7、n=16p5、n=64p4、n=2pq,其中p、q≥5为奇素数;所有奇数解为n=p、n=pαq,其中α≥1,p、q为奇素数.

By using the elementary number theory and combinational methods, the positive integer solutions of a function equation involving both of the Smarandache dual function and the Ω function is studied. All the exact posi- tive integer solutions are given for the equation, and it is proved that the even n satisfy the equation only if n=2^4＊3^30、n=2^5·3^12、n=8p^2、n=16p^5、n=64p^4、n=2pq , where p,q≥are both odd primes, and the odd n satisfy the equation only if n=p, n=p^＊q, where p, q are both odd primes, α≥1.