On multiplicatively perfect numbers (Q2710138)

Let \(n\) be a positive integer. Recall that \(n\) is called perfect if \(\sigma(n)= 2n\), superperfect if \(\sigma(\sigma(n))= 2n\), and \(k\)-perfect if \(\sigma(n)= kn\) (here, \(k>1\) is a fixed positive integer). In this paper, the author considers multiplicative analogues of such notions. Namely, for every positive integer \(n\) let \(T(n)\) stand for the product of all the divisors of \(n\). Then the author calls \(n\) multiplicatively perfect (or, \(m\)-perfect) if \(T(n)= n^2\), \(m\)-superperfect if \(T(T(n))= n^2\), and \(k\)-\(m\)-perfect if \(T(n)= n^k\). The author proves various results characterizing \(m\)-perfect, \(m\)-superperfect and \(k\)-\(m\)-perfect numbers for some values of \(k\). NEWLINENEWLINENEWLINEFor example, all the \(m\)-perfect numbers are the ones of the form \(pq\) or \(p^3\), where \(p\) and \(q\) are distinct primes, and there are no \(m\)-superperfect numbers (Theorem 2.1). Theorem 3.1 characterizes all the \(k\)-\(m\)-perfect numbers for \(k\leq 10\). For example, all 6-\(m\)-perfect numbers are of the form \(pqr^2\), or \(pq^5\), or \(p^{11}\), where \(p\), \(q\) and \(r\) are distinct primes. The author also finds all the \(k\)-\(m\)-superperfect numbers for \(k\leq 9\), \(k\neq 8\) (Theorem 4.1). The intersection between the (usual) perfect numbers and the \(m\)-perfect, or \(k\)-\(m\)-perfect, etc. numbers is also investigated. NEWLINENEWLINENEWLINEThe last section of the paper is a collection of inequalities involving the function \(T(n)\). For example, the normal order of \(\log\log T(n)\) is \((1+\log 2)\log\log n\). The proofs are elementary.