Upper bounds for unitary perfect numbers and unitary harmonic numbers. (English) Zbl 1141.11006

A positive integer \(d\) is said to be a unitary divisor of \(N\) if \(d\mid N\) and \((d, N/d)=1\). The unitary divisor function \(\sigma_j^{\ast}(N)\) is defined as the sum of the \(j\)th powers of the unitary divisors of \(N\). A positive integer \(N\) is said to be a unitary perfect number (UPN) if \(\sigma_1^{\ast}(N)=2N\), see M. V. Subbarao and L. J. Warren [Can. Math. Bull. 9, 147–153 (1966; Zbl 0139.26901)]. Five UPNs are known and it is open whether there exist other UPNs. A positive integer \(N\) is said to be a unitary harmonic number (UHN) if the harmonic mean of its unitary divisors
\[ H^{\ast}(N)={N\sigma_0^{\ast}(N)\over \sigma_1^{\ast}(N)} \]
is integral, see K. Nageswara Rao [Scripta Math. 28, 347–352 (1965; Zbl 0216.03802)] or P. Hagis jun. and G. Lord [Proc. Am. Math. Soc. 51, 1–7 (1975; Zbl 0309.10004)]. A large number of examples of UHNs are known. The present author presents the list of all UHNs with \(H^{\ast}(N)\leq 50\). It is open whether there exist infinitely many UHNs.
The main results of this paper are the following. If \(N\) is a UPN (resp. a UHN) with \(k\) distinct prime factors, then \(N<2^{2^k}\) (resp. \(N<(2^{2^k})^k\)).
For a survey of UPNs, UHNs and related numbers, see J. Sándor and B. Crstici [Handbook of number theory II. (Dordrecht): Kluwer Academic Publishers (2004; Zbl 1079.11001)].


11A25 Arithmetic functions; related numbers; inversion formulas
11Y70 Values of arithmetic functions; tables
Full Text: DOI Euclid


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