\[\sum_{i=1}^{n}\sum_{j=1}^{n}(i+j)^{t}f(\gcd(i,j))\gcd(i,j)\]\[\sum_{i=1}^{n}\sum_{j=1}^{n}(i+j)^{t}\mu(\gcd(i,j))^{2}\gcd(i,j)\]\[\sum_{k=1}^{n}k^{t+1}\sum_{i=1}^{\lfloor\frac{n}{k}\rfloor}\sum_{j=1}^{\lfloor\frac{n}{k}\rfloor}(