Suppose $k_n$ denotes either $\phi (n)$ or $\phi (p_n)$ $(n \ = \ 1,2,\cdots )$ where the polynomial $\phi$ maps the natural numbers to themselves and $p_k$ denotes the $k^{th}$ rational prime. Let $(\frac{r_n}{q_n})_{n=1}^{\infty}$ denote the sequence of convergents to a real number $x$ and define the the sequence of approximation constants $(\theta _n(x))_{n=1}^{\infty}$ by $$\theta _n(x) \ = \ q_n^2\left\vert x \ - \ \frac{r_n}{q_n}\right\vert . \eqno (n \ = \ 1,2, \cdots )$$ In this paper we study the behaviour of the sequence $(\theta _{k_n}(x))_{n=1}^{\infty}$ for almost all $x$ with respect to Lebesgue measure. In the special case where $k_n \ = \ n$ $(n \ = \ 1,2,\cdots )$ these results are due to W. Bosma, H. Jager and F. Wiedijk.