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

Goto, Takeshi

Rocky Mt. J. Math. 37, No. 5, 1557-1576 (2007).

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)].

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

Citations:

Zbl 0139.26901; Zbl 0216.03802; Zbl 0309.10004; Zbl 1079.11001

References:

[1]	R.P. Brent, G.L. Cohen and H.J.J. te Riele, Improved techniques for lower bounds for odd perfect numbers , Math. Comp. 57 (1991), 857-868. JSTOR: · Zbl 0736.11004
[2]	J.E.Z. Chein, An odd perfect number has at least \(8\) prime factors , Ph.D. thesis, Pennsylvania State University, 1979.
[3]	G.L. Cohen and R.M. Sorli, Harmonic seeds , Fibonacci Quart. 36 (1998), 386-390; Errata, Fibonacci Quart. 39 (2001), 4. · Zbl 0948.11004
[4]	R.J. Cook, Bounds for odd perfect numbers , in Number theory , CRM Proc. Lecture Notes 19 (1999), 67-71. · Zbl 0928.11003
[5]	L.E. Dickson, Finiteness of odd perfect and primitive abundant numbers with \(n\) distinct prime factors , Amer. J. Math. 35 (1913), 413-422. JSTOR: · JFM 44.0220.02
[6]	T. Goto and S. Shibata, All numbers whose positive divisors have integral harmonic mean up to \(300\) , Math. Comp. 73 (2004), 475-491. · Zbl 1094.11005
[7]	P. Hagis, Jr., Outline of a proof that every odd perfect number has at least eight prime factors , Math. Comp. 35 (1980), 1027-1032. JSTOR: · Zbl 0444.10004
[8]	P. Hagis, Jr. and G. Lord, Unitary harmonic numbers , Proc. Amer. Math. Soc. 51 (1975), 1-7. · Zbl 0309.10004
[9]	D.R. Heath-Brown, Odd perfect numbers , Math. Proc. Cambridge Philos. Soc. 115 (1994), 191-196. · Zbl 0805.11005
[10]	Pace P. Nielsen, An upper bound for odd perfect numbers , Integers 3 (2003), A14, 9 pp. (electronic). · Zbl 1085.11003
[11]	O. Ore, On the averages of the divisors of a number , Amer. Math. Monthly 55 (1948), 615-619. JSTOR: · Zbl 0031.10903
[12]	C. Pomerance, On a problem of Ore : Harmonic numbers , Notices Amer. Math. Soc. 20 (1973), A-648.
[13]	——–, Multiply perfect numbers, Mersenne primes, and effective computability , Math. Ann. 226 (1977), 195-206. · Zbl 0324.10001
[14]	M.V. Subbarao, Are there an infinity of unitary perfect numbers? Amer. Math. Monthly 77 (1970), 389-390. JSTOR:
[15]	M.V. Subbarao and L.J. Warren, Unitary perfect numbers , Canad. Math. Bull. 9 (1966), 147-153. · Zbl 0139.26901
[16]	C.R. Wall, The fifth unitary perfect number , Canad. Math. Bull. 18 (1975), 115-122. · Zbl 0312.10004
[17]	——–, New unitary perfect numbers have at least nine odd components , Fibonacci Quart. 26 (1988), 312-317. · Zbl 0657.10003