  The height $H(n)$ of $n$, introduced by Pillai in 1929, is the smallest
positive integer $i$ such that the $i$th iterate of Euler's totient function at
$n$ is $1$. H. N. Shapiro (1943) studied the structure of the set of all
numbers at a height. We state a formula for the height function due to Shapiro
and use it to list steps to generate numbers at any height. This turns out to
be a useful way to think of this construct. In particular, we extend some
results of Shapiro regarding the largest odd numbers at a height. We present
some theoretical and computational evidence to show that $H$ and its relatives
are closely related to the important functions of number theory, namely
$\pi(n)$ and the $n$th prime $p_n$. We conjecture formulas for $\pi(n)$ and
$p_n$ in terms of the height function.
