Coprime-factorization lower bound for a quadratic-form class number

Let $d$ be odd and have $k$ distinct prime divisors. Assigning each complete prime-power factor of $d$ to one of two factors gives $2^k$ ordered coprime factorizations $d=ac$. Interchanging $a$ and $c$ identifies pairs, and each unordered factorization gives a different reduced primitive form $[\min (a,c),0,\max (a,c)]$ of discriminant $-4d$. Hence