Another New Function

The totient function $\phi (n)$ or $phi(n)$, also called Euler's totient function, is defined as the number of positive integers $\le n$ that are relatively prime to (i.e., do not contain any factor in common with) $n$, where $1$ is counted as being relatively prime to all numbers. Since a number less than or equal to and relatively prime to a given number is called a totative the totient function $\phi (n)$ can be simply defined as the number of totatives of $n$. For example, there are eight totatives of $24$ ($1$, $5$, $7$, $11$, $13$, $17$, $19$ and $23$), so \varphi(24) = 8.

The depth of $phi$ value of a number is denoted by the number of steps required before it reaches $1$. An example will make it very clear.

• \varphi(13) = 12 . . . step $1$

• \varphi(12) = 4 . . . step $2$

• \varphi(4) = 2 . . . step $3$

• \varphi(2) = 1 . . . step $4$

So the depth of $phi(13)$ is $4$. We name this function as $depthphi$. So we can write depthphi(13) = 4. The sum of $depthphi$ function ($SODF$) takes two integers as parameter and its definition is given below:

Given the value of $m$ and $n$ your job is to find the value of SODF(m, n).