HMMT 2004 T16

Now suppose again that $n$ is odd. Prove that

$$\sigma(1)\left\lfloor\log _{2} n\right\rfloor+\sigma(3)\left\lfloor\log _{2}(n / 3)\right\rfloor+\sigma(5)\left\lfloor\log _{2}(n / 5)\right\rfloor+\cdots+\sigma(n)\left\lfloor\log _{2}(n / n)\right\rfloor<n^{2} / 8$$