Multiple questions have been submitted regarding the sum of the reciprocal of primorials, i.e.
$$ S_P=\sum_{n=1}^{\infty}P_n = \frac{1}{2} + \frac{1}{2\cdot3} + \frac{1}{2\cdot3\cdot5} + \ldots \approx 0.7052301717918\ldots $$
I got curious what this would look like if we define an "anti-primorial" $A_n$ to be the product of the $n$ first non-primes, and calculate the sum of their reciprocals:
$$ S_A=\sum_{n=1}^{\infty}A_n = \frac{1}{4} + \frac{1}{4\cdot6} + \frac{1}{4\cdot6\cdot8} + \ldots \approx 0.29751676555\ldots $$
The first thing that jumps to the eye is that the sum of these 2 values is quite close to 1 (or 2, if you include 1 in the "anti-primorials"):
$$ S_P + S_A \approx 1.0027469\ldots $$
Now this may well be a "coincidence" due to the first few biggest factors happening to have the right values, but I was wondering if perhaps someone would have some deeper insight on why this pattern might arise. Thanks.