Let $\phi_k(n)$ be the number of integers $m$ in $1\le m\le n$ for which $\gcd(m,n) = k$. Then $\phi_1(n) =\varphi(n)$, the standard totient function.

This function arises in the analysis of the structure of the cyclic group of order $n$: the number of elements of $Z_n$ that have order $k$ is $\phi_k(n)$. It similarly arises in the analysis of the dihedral groups, which is where I ran into it. So I imagine that it has a standard name.

It is easy enough to calculate, but I would like to research it and I don't know what it is called.

  • 2
    $\begingroup$ Wouldn't we just have $\phi_k(n)=\phi(\frac{n}k)$? I would think it wouldn't be named if that is indeed the case. $\endgroup$ – Milo Brandt May 14 '15 at 3:20
  • $\begingroup$ Yep, I was just coming back to say that. Thanks! $\endgroup$ – MJD May 14 '15 at 3:30

This is very easy; it's apparent that if $\phi_k(n)=0$ if $k$ does not divide $n$, and if it does divide $n$, then $$\phi_k(n) = \varphi\left(\frac nk\right)$$ because $\gcd(ka, kb) = k\cdot\gcd(a, b)$. This gives $\phi_1 = \varphi$ as desired.

So it's probably too simple to have a separate name.


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