I was curious about methods of finding primes. One method is to keep a list of the first $n$ primes, then check if the input modulo any prime less than its square root is zero. If none of those primes is a divisor, check all odd numbers between the greatest prime and the square root of the input for a factor.
Assuming the input is a random integer, I figured I could see how often this method finds a factor among the first $n$ primes.
Let $P_n$ be the set of the first $n$ primes. The asymptotic probability that a random input will have a factor among the first $n$ primes is:
$$k_n=1+\sum_{A\subseteq P_n}\dfrac{(-1)^{|A|+1}}{\displaystyle \prod_{a\in A}a}$$
Calculating the first few of these:
$$k_2=\dfrac{1}{2} \\ k_3=\dfrac{2}{3} \\ k_4=\dfrac{11}{15} \\ k_5=\dfrac{27}{35} \\ k_6=\dfrac{61}{77}$$
I'm sure this sequence is known, but it is not an integer sequence, and it was not found on Wolframalpha. Does this sequence have a name? I'm not sure how to look it up to find out more about it.