Jordan's totient function ($J_k(n)$)

For positive integers $k,n$, Jordan's totient function $J_k(n)$ counts tuples $(a_1,\dots ,a_k)$ of residue classes modulo $n$ for which $\gcd (a_1,\dots ,a_k,n)=1$. The coordinates together must avoid a common prime divisor of $n$; they need not each be units.

For each prime factor $p$ of $n$, exactly $(n/p{)}^k$ tuples have all coordinates divisible by $p$. The inclusion-exclusion principle, applied to simultaneous divisibility by the distinct prime factors, therefore givesIn particular $J_1(n)=\phi (n)$ is Euler's totient function. For $k=2,n=6$, the formula gives $J_2(6)=36(1-1/4)(1-1/9)=24$. The pair $(2,3)$ is counted because $\gcd (2,3,6)=1$, even though neither coordinate is coprime to six.