Unitary divisor

A divisor $d$ of a positive integer $n$ is unitary when $\gcd (d,n/d)=1$. Each prime power in the prime factorization of $n$ must therefore be included in $d$ completely or omitted completely. Consequently, the sum of the $k$-th powers of unitary divisors is ${\prod }_{p^e\parallel n}(1+p^{ek})$.

A unitary divisor $d$ of a positive integer $n$ satisfies $\gcd (d,n/d)=1$. The greatest common unitary divisor of $a,b$ is the largest integer that is a unitary divisor of both. In their prime factorizations, it includes precisely those whole prime powers whose positive exponents agree in $a$ and $b$.

For $a=12=2^2\cdot 3$ and $b=18=2\cdot 3^2$, it is $1$, although the ordinary greatest common divisor is $6$. For $12$ and $20=2^2\cdot 5$, it is $4$. Requiring unitary divisibility on both inputs is essential: merely being a unitary divisor of their ordinary greatest common divisor is a weaker condition.