一个包含Smarandache对偶函数的方程

An Equation Involving the Smarandache Dual Function

利用初等数论及组合方法讨论了一个包含Smarandache对偶函数的方程∑(d|n)1/(S*(d))=3Ω(n)的可解性.得到了其所有正整数解的具体形式,即方程所有奇数解为n=p3q 5(p,q为奇素数),所有偶数解为n=28·3114,n=210·336,n=216·318,n=2αp2,n=2pqr(α>1,p,q,r为大于3的奇素数).

The elementary number theory and combinational methods are used to study the solvability of a function equation ∑d｜n1/Sn（d）=3Ω（n）which involves the Smarandache dual function. All the exact positive integer solutions are given for the equation. All the odd solutions of the equation are n = p3 qS, where p,q are both odd primes; all the even solutions are n=28 · 3^114 , n=2^10 · 3^36 , n=2^16 · 3^18 , n=2ap2 , and n=2pqr, where a 〉 1, p,q,r are all odd primes greater than 3.