Here is the abstract:

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 provide a formula for the height function thereby extending a result of Shapiro. We list steps to generate numbers of any height which 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.Here is a link to a preprint of the paper.

Prime number conjectures from the Shapiro class structure (with Hartosh Singh Bal), 17 pp.

UPDATE (Dec 15, 2019): The paper has been accepted to appear in INTEGERS: Electronic Journal of Combinatorial Number Theory

From left to right: Sonit, Hartosh, me, Punya in 1983 or so

See here for my collaboration with Punya.