Suppose \(k_n\) denotes either \(\phi(n)\) or \(\phi(p_n)\) (\(n=1,2,\dots\)), where the polynomial \(\phi\) maps the natural numbers to themselves, and \(p_k\) denote the \(k^{th}\) rational prime. Also let \(({r_n\over q_n})_{n=1}^\infty\) denote the sequence of convergents to a real number \(x\) and let \((c_n(x))_{n=1}^\infty\) be the corresponding sequence of partial quotients for the nearest integer continued fraction expansion. Define the sequence of approximation constants \((\theta_n(x))_{n=1}^\infty\) by \[ \theta_n(x)=q_n^2\left| x-{r_n\over q_n}\right| \quad(n=1,2,\dots). \] In this paper, the author studies the behaviour of the sequence \((\theta_{k_n}(x))_{n=1}^\infty\) and \((c_{k_n}(x))_{n=1}^\infty\) for almost all \(x\) with respect to the Lebesgue measure. In the special case where \(k_n=n\) (\(n=1,2,\dots\)) these results are known and due to H. Jager [Indag. Math. 48, 61-69 (1986; Zbl 0588.10061)] and G. J. Rieger [J. Reine Angew. Math. 310, 171-181 (1979; Zbl 0409.10038)] and others.