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

We prove explicit asymptotic formulae for some functions used in sieve methods and show that there exists no odd multiperfect number of abundancy four whose squared part is cubefree.
Comment: 17 pages (The proof of Lemma 2.3 is refined and Theorem 3.1 is improved, freeing us from checking exceptional primes), the author's final version. Appeared in International J. Number Theory (Online Ready)