I was looking at the totient function, and I found the property that $pq-\phi(pq)$, where $p$ and $q$ are primes, can give you any odd number. It is trivial to prove this for primes, because $p=p^2-\phi$(p^2).
Then I define the $N$ to be the $p_1\cdot p_2\cdot \cdots \cdot p_n$, and then I substitute $p=p^2-\phi(p^2)$, giving $$ N=(p_1^2-\phi(p1^2))\cdot(p_2^2-\phi(p_2^2))\cdot \cdots \cdot(p_n^2-\phi(p_n^2)). $$ I have started to expand this, but it is taking a lot of work and if there is an easier way to do this I would prefer to avoid brute force.
What I'm asking is how could you expand the above expression, and keep it in terms of the totient function? Sorry for my ugly formatting, I don't know LaTeX, and for the love of god please don't write that $N$ is the expansion. That much I already know.