Erdős Problem #33 — Minimal counting function of complements to the squares

Let $b_1 < b_2 < \ldots$ be an infinite sequence of integers, so that every integer is of the form $b + k^2$. Denote by $B(x)$ the number $b$-s not exceeding $x$. [...] I can not determine the smallest possible value of $\limsup \frac{B(x)}{x^{\frac{1}{2}}}$. Clearly $B(x) \geqslant x^{\frac{1}{2}}$ holds, but I can not prove that $\liminf \frac{B(x)}{x^{\frac{1}{2}}} > 1$.