Twos are all you need

 Published on Saturday, 28th March 2020, 01:00 pm; Solved by 261;
Difficulty rating: 50%

Problem 708

A positive integer, $n$, is factorised into prime factors. We define $f(n)$ to be the product when each prime factor is replaced with $2$. In addition we define $f(1)=1$.

For example, $90 = 2\times 3\times 3\times 5$, then replacing the primes, $2\times 2\times 2\times 2 = 16$, hence $f(90) = 16$.

Let $\displaystyle S(N)=\sum_{n=1}^{N} f(n)$. You are given $S(10^8)=9613563919$.

Find $S(10^{14})$.