On $\boldsymbol \phi$-amicable pairs

Let $\phi (n)$ denote Euler's totient function, i.e., the number of positive integers $<n$ and prime to $n$. We study pairs of positive integers $(a_{0},a_{1})$ with $a_{0}\le a_{1}$ such that $\phi (a_{0})=\phi (a_{1})=(a_{0}+a_{1})/k$ for some integer $k\ge 1$. We call these numbers $\phi$-amicable pairs with multiplier $k$, analogously to Carmichael's multiply amicable pairs for the $\sigma$-function (which sums all the divisors of $n$).

We have computed all the $\phi$-amicable pairs with larger member $\le 10^{9}$ and found $812$ pairs for which the greatest common divisor is squarefree. With any such pair infinitely many other $\phi$-amicable pairs can be associated. Among these $812$ pairs there are $499$ so-called primitive $\phi$-amicable pairs. We present a table of the $58$ primitive $\phi$-amicable pairs for which the larger member does not exceed $10^{6}$. Next, $\phi$-amicable pairs with a given prime structure are studied. It is proved that a relatively prime $\phi$-amicable pair has at least twelve distinct prime factors and that, with the exception of the pair $(4,6)$, if one member of a $\phi$-amicable pair has two distinct prime factors, then the other has at least four distinct prime factors. Finally, analogies with construction methods for the classical amicable numbers are shown; application of these methods yields another 79 primitive $\phi$-amicable pairs with larger member $>10^{9}$, the largest pair consisting of two 46-digit numbers.