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On \(k\)-imperfect numbers. (English) Zbl 1175.11004

A positive integer \(n\) is said to be \(k\)-imperfect if \(k\rho(n)=n\) for some integer \(k\geq2\), where \(\rho\) is the multiplicative arithmetic function defined by \(\rho(p^\alpha)=p^\alpha-p^{\alpha-1}+p^{\alpha-2}-\cdots+(-1)^{\alpha}\) for a prime power \(p^\alpha\). Unsolved problems relating to \(k\)-imperfect numbers and the \(\rho\)-function are similar to those relating to \(k\)-perfect numbers and the \(\sigma\)-function. Denoting \(H(n)=n/\rho(n)\) we can say that \(n\) is a \(k\)-perfect number if \(H(n)=k\).
G. Martin introduced the function \(\rho\) (using the notation \(\tilde\sigma\)) at the 1999 Western Number Theory Conference and raised three questions:
(1) Are there \(k\)-imperfect numbers with \(k\geq 4\)?
(2) Are there infinitely many \(k\)-imperfect numbers?
(3) Are there any odd \(3\)-imperfect numbers?
See Richard K. Guy [Unsolved problems in number theory. 3rd ed. Problem Books in Mathematics. New York, NY: Springer-Verlag (2004; Zbl 1058.11001)].
D. E. Ianucci [Integers 6, Paper A41, 13 p. (2006; Zbl 1114.11004)] gave several necessary conditions for odd \(3\)-imperfect numbers and listed all \(k\)-imperfect numbers up to \(10^9\); these \(k\)-imperfect numbers are all even. If we can find an odd \(k\)-imperfect number, then Question (1) can be answered. In fact, if \(n\) is an odd \(k\)-imperfect number, then \(H(2n)=2k\geq 4\).
The present authors prove that every odd \(k\)-imperfect number greater than \(1\) must be divisible by a prime greater than \(10^2\) and give all \(k\)-imperfect numbers less than \(2^{32}=4\,294\,967\,296\). They also give several necessary conditions for the existence of odd \(k\)-imperfect numbers.
For a general account of perfect numbers, see J. Sándor and B. Crstici [Handbook of number theory II. Dordrecht: Kluwer Academic Publishers (2004; Zbl 1079.11001)].

MSC:

11A25 Arithmetic functions; related numbers; inversion formulas
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