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**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)].

\[ H^{\ast}(N)={N\sigma_0^{\ast}(N)\over \sigma_1^{\ast}(N)} \]

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)].

Reviewer: Pentti Haukkanen (Tampere)

### MSC:

11A25 | Arithmetic functions; related numbers; inversion formulas |

11Y70 | Values of arithmetic functions; tables |

### Keywords:

unitary divisor; unitary divisor function; unitary perfect number; unitary harmonic number; bound
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\textit{T. Goto}, Rocky Mt. J. Math. 37, No. 5, 1557--1576 (2007; Zbl 1141.11006)

### Online Encyclopedia of Integer Sequences:

Unitary perfect numbers: numbers k such that usigma(k) - k = k.Unitary harmonic numbers (those for which the unitary harmonic mean is an integer).

### References:

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