Let \(\mathcal{C}\) denote the knot concordance group, which was originally defined by R. H. Fox and J. W. Milnor [Osaka J. Math. 3, 257–267 (1966; Zbl 0146.45501)]. T. D. Cochran et al. [Ann. Math. (2) 157, No. 2, 433–519 (2003; Zbl 1044.57001)] produced a filtration of \(\mathcal{C}\) indexed by half integers, namely \[ \dots \leq \mathcal{F}_{2} \leq \mathcal{F}_{1.5} \leq \mathcal{F}_{1} \leq \mathcal{F}_{0.5} \leq \mathcal{F}_{0} \leq \mathcal{C} \] A knot in \(\mathcal{F}_{n}\) is called \(n\)-solvable. It is known that \(\mathcal{F}_{0}\) consists of knots with Arf-invariant zero, and \(\mathcal{F}_{0.5}\) consists of algebraically slice knots. Remember that a knot is algebraically slice if there exists a Seifert surface \(F\) for \(K\) such that the Seifert form vanishes on a half-rank submodule of \(H_1(F;\mathbb{Z})\). T. D. Cochran et al. [Math. Ann. 351, No. 2, 443–508 (2011; Zbl 1234.57004)] showed that for all \(n\in \mathbb{N}\), \(\mathcal{F}_{n}/\mathcal{F}_{n.5}\) contains an infinite rank free abelian subgroup. On the other hand, for the group \(\mathcal{F}_{n.5}/\mathcal{F}_{n+1}\) not much is known. Results of this paper suggest that \(\mathcal{F}_{0.5}=\mathcal{F}_{1}\). It is proved that if a knot \(K\) has genus 1 and is algebraically slice, then \(K\) is \(1\)-solvable. It is also shown that if \(K\) is algebraically slice and it has genus \(2\), \(3\), or \(n\) and certain conditions are satisfied in each case, then \(K\) is \(1\)-solvable. Examples are given of knots satisfying the conditions of the results.