I stumbled upon the following problem, while trying to come up with a recreational math question.
Let $n$ be a positive integer with factorization $n=2^a\prod_{i=1}^{k}p_i^{e_i}$
Define the arithmetic function $$f(n)=2^{\left\lfloor \frac{a}{2} \right\rfloor}\prod_{i=1}^{k}p_i^{\left\lfloor\frac{e_i+1}{2}\right\rfloor}$$ For example, $n=120=2^3\cdot 3\cdot 5$, so $f(120)=2^1\cdot 3^1 \cdot 5^1=30$.
I need to evaluate the sum $$S(N)=\sum_{n=1}^{N}\left\lfloor\frac{N}{f(n)}\right\rfloor$$ When $N$ becomes large ($>10^8$ say), factorizing each number is out of question. I wonder if there is some möbius inversion kind of trick to get the sum for very large $N$, say $10^{14}$.