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On quasi-imperfect numbers with at most four distinct prime divisors. (English) Zbl 1483.11006

Summary: Let \(\rho\) be a multiplicative arithmetic function defined by \(\rho (p^{\alpha })=p^{\alpha }-p^{\alpha -1}+p^{\alpha -2}-\cdots +(-1)^{\alpha }\) for a prime power \(p^{\alpha }\). For a positive integer \(n\), we call \(n\) a quasi-imperfect number if \(2\rho (n)=n+1\). In this paper, we show that there are only four quasi-imperfect numbers with at most four distinct prime divisors. We also pose some conjectures for further research.

MSC:

11A25 Arithmetic functions; related numbers; inversion formulas
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