A note on amicable numbers and their variations (Q1677548)

Let \(\sigma(n)\) denote the sum of the positive divisors of \(n\). For a fixed positive integer \(a\), if there exists a positive integer \(b\) (\(\neq a\)) such that \(\sigma(a)=\sigma(b)=a+b\), then \((a, b)\) is said to be an amicable pair. If there does not exist such a positive integer \(b\), then \(a\) is an anti-sociable number. Similarly, if \(a\) and \(b\) satisfy \(\sigma(a)=\sigma(b)=a+b+1\) or \(\sigma(a)=\sigma(b)=a+b-1\), then \((a,b)\) is a quasi-amicable or an augmented amicable pair, respectively. \smallskip It is not known whether any amicable pairs \((a, b)\) exists with \(a\) and \(b\) opposite parity, or with \(\gcd(a,b)=1\). The present authors show that if \(a\) and \(b\) are positive integers such that \(4 \mid a\) and \(\gcd(a, b)=1\), then \((a, b)\) is not an amicable pair. \smallskip For any positive integer \(n\) with \(n>1\), let \(p(n)\) denote the least prime divisor of \(n\). It is known that, for some special positive integers \(a\), if \(p(a)\) is large enough, then \(a\) must be an anti-sociable number. The present authors prove that if \(a>10^8\) and \(p(a)>2\log_2{a}+1\), then there does not exist a positive integer \(b\) such that the pair \((a, b)\) is amicable, quasi-amicable or augmented amicable.