Question 154
Let $d(k)$ denote the number of positive integer divisors of $k$. For example, $d(6)=4$ since $6$ has $4$ positive divisors, namely, $1, 2, 3$, and $6$. Prove that for all positive integers $n$,
$$
d(1)+d(3)+d(5)+\cdots+d(2n-1) \leq d(2)+d(4)+d(6)+\cdots+d(2n)
$$
$$
d(1)+d(3)+d(5)+\cdots+d(2n-1) \leq d(2)+d(4)+d(6)+\cdots+d(2n)
$$