Let the set $ S_{n}$ = {$ (x,y):x,y \in \mathbb{O}$ } such that $ x+2=y$ and $ y$ is less than or equal to odd integer $ n$ and $ \mathbb{O}$ is set of odd integers > 1.

Let us define the function $ f(n) = |S_{n}|$ that counts the number of pairs in $ S_{n}$ .

Examples: $ S_{9} =\{(3,5),(5,7),(7,9)\}$ and $ f(9) = 3$ .

$ S_{13} = \{(3,5), (5,7), (7,9), (9,11),(11,13)\}$ and $ f(13) = 5$

$ S_{35} = \{(3,5), (5,7), (7,9), (9,11), (11,13), (13,15), (15,17), (17,19), (19,21), (21,23), (23,25), (25,27), (27,29), (29,31),(31,33),(33,35)\}$

and $ f(35) = 16$

It turns out that $ f(n) = ((n-3)/2)$ for odd integers $ n$ .

Let us define another function $ g(n)$ that counts the number of pairs $ (x,y)$ such that either $ x$ or $ y$ is divisible by 5 and not divisible by 3, and $ x \neq 5$ and $ y\neq 5$ .

Example: $ g(35) = 3$ because there are 3 pairs where $ x$ or $ y$ is divisible by 5 and not 3, and $ x$ or $ y$ is not equal to 5. They are (23,25), (25,27) and (33,35).

Note, the pairs (3,5) and (5,7) are not included since they contain 5, and the pairs (13,15) and (15,17) are not included because 15 is also divisible by 3.

What is the formula for $ g(n)$ in terms of $ f(n)$ ?

What is the formula for $ g(n)$ in terms of $ f(n)$ for limit $ n \to\infty$ ?