  The prime-counting function $\pi(x)$ which returns the number of primes
smaller or equal to a given number is a topic of interest in number theory. An
algorithm based on a cyclic group isomorphic to $Z/nZ$, the so-called
$Z$-functions, was proposed in view to outperform its pieers. The approach
suggests a time complexity $\mathcal{O}(x^{1/2})$ in agreement with optimality
of a 2-D squared adaptive-recursive algorithm. The present work is a
presentation of various approaches as ascending factorization, the main
sequence of odd numbers and partial sequences, T-series, counting function of
prime composites, $Z$-modular forms and combinatorial aspects.
