For each positive integer $k$, let $A(k)$ be the number of odd divisors of $k$ in the interval $[1,\sqrt{2k}).$ Evaluate $$\sum_{k=1}^{\infty}(-1)^{k-1}\frac{A(k)}{k}.$$ The original answer is in the format $\frac{m\pi^j}{n}$, please provide the value of m + n + j.

Let's think step by step and output the final answer within \boxed{}.
user