Explicit sieve estimates and nonexistence of odd multiperfect numbers of a certain form (Q6591611)

A positive integer \(N\) is called perfect if the sum of divisors of \(N\) is equal to \(2N\). More generally, a positive integer \(N\) is \(k\)-perfect if the sum of divisors of \(N\) is equal to \(kN\). Euler showed that an odd perfect number must be of the form \(N=q_r^{\alpha} q_1^{2\beta_1}\cdots q_{r-1}^{2\beta_{r-1}}\) where \(q_1, \ldots, q_r\) are distinct odd primes and \(\alpha, \beta_1, \ldots, \beta_{r-1}\) are positive integers with \(q_r\equiv \alpha\equiv 1\pmod{4}\). In the paper under review, the author considers odd perfect numbers of the special form\N\[\NN=q_r^{\alpha} (q_1 q_2 \cdots q_{r-1})^{2\beta},\N\]\Nand shows validity of the bound\N\[\Nr<\left(2\beta+\frac{\log\log \beta+1.24351}{\log 3}\right)(\beta+3)+4\N\]\Nfor \(\beta\geq 8\) and \(r\leq 14\), \(30\), \(56\), \(90\), \(132\), \(182\), and \(240\) for \(\beta=1\), \(2\), \(3\), \(4\), \(5\), \(6\), and \(7\), respectively. Also, he shows that no integer of the form \(N=3^\alpha (q_1 q_2 \cdots q_{r-1})^2\) with \(3, q_1, q_2, \ldots, q_{r-1}\) distinct odd primes is a \(4\)-perfect number. The proof is based on Selberg's sieve with explicit estimates.