In this question Larry Freeman showed that $\prod_1^n(p_i-2)$ reports the number of pairs of numbers $x,(x+2)$ such that $x<p_n\# \wedge \gcd (x(x+2),p_n\#)=1$.
This bears a striking resemblance to Euler's totient function applied to primorials, which states $\phi(p_n\#)=\prod_1^n(p_i-1)$ and reports the number of numbers $x$ such that $x<p_n\# \wedge \gcd (x,p_n\#)=1$. More generally, for $x=\prod(p_i^{a_i})$, the totient $\phi(x)=\prod(p_i^{a_i-1})(p_i-1)$.
I was curious to see whether Freeman's observation could be generalized, and looking at a small number of examples, I formulated a totient-like function which I denote $\phi_2(x)$: $$\phi_2(2)=1\\ \phi_2(p)=(p-2);\ p>2\in \mathbb P\\ \phi_2(2^a)=2^{a-1}\\ \phi_2(p^a)=p^{a-1}(p-2)$$ The function is multiplicative: $$\phi_2(m)\cdot \phi_2(n)=\phi_2(mn) \text{ when }\gcd(m,n)=1$$ This totient-like function seems to report the number of numbers $n$ smaller than $x$ such that $\gcd (n(n+2),x)=1$. Note that $n+2$ need not be smaller than $x$; the pair $(x-1),(x+1)$ is counted by this function.
Questions: 1. Have I reinvented a well known wheel? I could not find anything about a function of this sort in a quick search of Wikipedia topics. Any references or citations would be appreciated. 2. The properties of the function I have observed are based on counting/computation for a small number of examples. Is it possible to prove that $\phi_2(x)$ in fact reports $n<x$ such that $\gcd (n(n+2),x)=1$?
