The unitary amicable pairs to \(10^ 8\) (Q1805145)

A unitary amicable pair is a pair of positive integers \((m,n)\) for which \(\sigma^* (m)= \sigma^* (n)= m+n\), where \(\sigma^*\) is the sum of unitary divisor functions. This paper presents an exhaustive list of the 185 unitary amicable pairs with smaller member \(\leq 10^8\), and one new unitary aliquot cycle of length 4, constructed from another aliquot cycle which is based on the sum of divisor functions, and which was found recently by A. Flammenkamp. The list was computed on a NeXT station with the help of Mathematica. No attempt is made to mark related pairs in the list. For example, if \((m, n)\) is a unitary amicable pair and if 12 is a unitary divisor of both \(m\) and \(n\), then replacing the common factor 12 by 18 yields another unitary amicable pair. Another such replacement pair is \(3^3\), \(3^4 41\). Examples are pairs \(\#\# 5\) and 7, and 142 and 179. Another rule which identifies related unitary amicable pairs is the following, which is analogous to a similar rule given by the reviewer for ordinary amicable pairs [Math. Comput. 47, 361-368 (1986; Zbl 0598.10012), Theorem 2 on page 366]: Let \((au, ap)\) be a given unitary amicable pair, where \(p\) is a prime with \(\text{gcd} (a,p)=1\) and let \(C= (p+1) (p+u)\). Write \(C= D_1 D_2\) with \(0< D_1< D_2\). If the three integers \(r=p+ D_1\), \(s=p+ D_2\), and \(q= u+r +s\) are primes not dividing \(a\), then \((auq, ars)\) is also a unitary amicable pair. Pairs related by this rule are \(\#\# 4\) and 121, and 6 and 160. [Reviewer's remark: The reviewer has repeated the computations in the interval \([9\times 10^7, 10^8]\) and found complete agreement with the list].