Do you know a name and expression of this function of Lehmer? In the paper "A new calculus of numerical functions." (D.H. Lehmer, 1931) Lehmer defines a function d as follows.
The function d(i,n) may be defined as follows: d(i,n)=0, if i is not a divisor of n. Otherwise d(i,n) is the largest divisor d of i for which n/d is prime to d.
I would like to know if, later, this function was named in anyway. Do you know an equivalent expression of this function in more generally known arithmetical functions?
UPDATE:
There is a theorem that converts an LCM sum to a Divisor sum.
$$\sum_{[ab]=n}f(a)=\sum_{\delta/n}f(\delta)\tau(d(\delta,n))$$
where $\tau$ is the common divisor count function, [ab] are all 2-tuples with LCM[a,b]=n, d the function as mentioned above.
 A: I believe the function $d(i,n)$ can be computed rather efficiently for large magnitudes of $i$ and $n$ without knowing the factors/divisors of $i$ or $n$ as follows. The formulas below are expressed in the Wolfram language and were implemented and tested in Mathematica. Note the $f(i,n)$ function defined in formula (2) below returns the final value of the local variable $y$.

(1) $\ d(\text{i$\_$},\text{n$\_$})\text{:=}\left\{
\begin{array}{cc}
 f(i,n) & i|n \\
 0 & \text{True} \\
\end{array}\right.$
(2) $ f(\text{i$\_$},\text{n$\_$})\text{:=}\text{Module}\left[\left\{y=i,g=\gcd \left(\frac{n}{i},i\right)\right\},\text{While}\left[g\neq 1,y=\frac{y}{g};g=\gcd \left(\frac{n}{y},y\right)\right];y\right]$

The $f(i,n)$ function defined in formula (2) above can also be implemented recursively as follows.

(3) $\ f(\text{i$\_$},\text{n$\_$})\text{:=}\text{Module}\left[\left\{g=\gcd \left(\frac{n}{i},i\right)\right\},
\left\{\begin{array}{cc}
 f\left(\frac{i}{g},n\right) & g\neq 1 \\
 i & \text{True} \\
\end{array}\right.
\right]$

Here's a conjectured relationship between the function $d(i,n)$ and OEIS Entry A165430: Table T(n,m) read by rows: the greatest common unitary divisor of n and m, n>=1, 1<=m<=n.

(4) $\quad d(i,n)\ne 0\implies d(i,n)=T(i,n)$
