Conjecture. The following arithmetic function is never zero, for any $t \in \Bbb{R}$, and $t \geq 5$: $$ g(t) := \sum_{d \mid p_n\#}(-1)^{\omega(d)}\left\lfloor\frac{t}{d}\right\rfloor|G_d| $$ where $G_d = \{ x \in \Bbb{Z}/d : x^2 = 1\pmod d\}$, and where $n(x) := \pi(\sqrt{x + 1})$ varies with $x$.
What it's related to is the prime-counter function $\pi(x)$. If you were to replace each of the groups of 2nd roots of unity with the trivial group then you'll recover:
$$ \sum_{d \mid p_n\#} (-1)^{\omega(d)}\left\lfloor\frac{t}{d}\right \rfloor = \pi(t) - \pi(\sqrt{t+ 1}) $$
This is well-known and even on the Wikipedia page for the prime counter. This is a generalization of that function, clearly. What it counts, I'm not sure. It doesn't count quite the twin prime averages. However, it does show up in that context.