On Φ–amicable Pairs

Let φ(n) denote Euler’s totient function, i.e., the number of positive integers < n and prime to n. We study pairs of positive integers (a0, a1) with a0 ≤ a1 such that φ(a0) = φ(a1) = (a0 + a1)/k for some integer k ≥ 1. We call these numbers φ–amicable pairs with multiplier k, analogously to Carmichael’s multiply amicable pairs for the σ–function (which sums all the divisors of n). We have computed all the φ–amicable pairs with larger member ≤ 109 and found 812 pairs for which the greatest common divisor is squarefree. With any such pair infinitely many other φ–amicable pairs can be associated. Among these 812 pairs there are 499 so-called primitive φ–amicable pairs. We present a table of the 58 primitive φ–amicable pairs for which the larger member does not exceed 106. Next, φ–amicable pairs with a given prime structure are studied. It is proved that a relatively prime φ–amicable pair has at least twelve distinct prime factors and that, with the exception of the pair (4, 6), if one member of a φ–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 φ–amicable pairs with larger member > 109, the largest pair consisting of two 46-digit numbers.