On $ \phi $ -amicable pairs (with appendix)

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 $k\ge1$. We call these numbers $\phi$--{\it 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 $\le10\^9$ 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. We present a table of the $58$ so-called 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 (a.o.) that a relatively prime $\phi$--amicable pair has at least twelve distinct prime factors and that, with the exception of the pair $(2^2,2\cdot 3)$, if one member of a $\phi$--amicable pair has two distinct prime factors, then the other has at least four distinct prime factors. 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.