__I. Counting Arithmetic Progressions__

__I. Counting Arithmetic Progressions__

Given an arbitrary increasing sequence of positive integers $S$ there is associated with it, a counting function $\alpha_S(n)$ which determines the number of arithmetic progressions

^{[1]}up to $s_n \in S$. The counting function for the positive integers, and consequently all infinite arithmetic progressions, is given by the formula listed under A330285 (OEIS)^{[2]}:\[ \sum_{k=1}^n \sum_{j=1}^k \Big{\lfloor} \frac{k-1}{j+1} \Big{\rfloor} \ . \]

We may further generalize from this "AP" counting function, $\alpha_{\mathbb{Z}^+}(n)$, all other functions which count progressions over integer sequences via the equation

\[ \alpha_S(n) = \sum_{k=1}^n \alpha_{\mathbb{Z}^+}(k) \pi (n,k) \]

where $\pi (n,k)$ is the number of

*primitive arithmetic progressions*of length $k$, up to the $n$-th element of $S$ .