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Rules for constructing hyperperfect numbers. (English) Zbl 0531.10005

In what follows m and n denote natural numbers while k is a non-negative integer. \(\sigma\) (m) and \(\omega\) (m) denote, respectively, the sum of the positive divisors and the number of distinct prime factors of m. If \(M_ n=\{m| \quad m=1+n(\sigma(m)-m-1)\}\) and \(_ kM_ n=\{m| \quad m\in M_ n\quad\text{and}\quad \omega(m)=k\}\) then the set of hyperperfect numbers is defined by \(M=\cup^{\infty}_{n=1}M_ n\), while the set of hyperperfect numbers with exactly k prime factors is given by \({}_ kM=\cup^{\infty}_{n=1}{_ kM_ n}\). We also define the sets \(M^*=\cup^{\infty}_{n=1}M^*_ n\) where \(M^*_ n=\{m| \quad m=1+n(\sigma(m)-m)\},\) and \({}_ kM^*=\cup^{\infty}_{n=1}{_ kM^*_ n}\) where \(_ kM^*_ n=\{m| \quad m\in M^*_ n\quad\text{and}\quad \omega(m)=k\}.\)
Hyperperfect numbers were first studied by D. Minoli and R. Bear [Pi Mu Epsilon J. 6, 153-157 (1975; Zbl 0319.10014)]. There are 71 elements of M below \(10^ 7\), 70 of which are in \({}_ 2M\). It is easy to show that \({}_ 1M=\emptyset\), and for some time it was thought that \({}_ kM=\emptyset\) if \(k>2\). This was shown to be false by the author [Math. Comput. 36, 297-298 (1981; Zbl 0452.10005)], and in the present paper rules are given by which elements of \(_{(k+2)}M_ n\) and \(_{(k+1)}M_ n\) can be constructed from elements of \({}_ kM^*_ n\). Since a rule is also given for constructing elements of \({}_ kM^*_ n\) from elements of \(_{(k-2)}M^*_ n\) and since \({}_ 0M^*_ n=\{1\}\) for all n while \(_ 1M^*_ n=\{(n+1)^{\alpha};\quad \alpha =1,2,3,...\}\) or \(\emptyset\) according as \(n+1\) is prime or composite it follows that, if one is willing to expend the computer time, elements of \({}_ kM\) for any \(k\geq 2\) may be sought in a systematic manner. Using his rules the author has found 165 elements of \({}_ 3M\), seven elements of \({}_ 4M\) and one element of \({}_ 5M\). A generalization of hyperperfect numbers to “hypercycles” is described in the final section of this paper.
Reviewer: P.Hagis

MSC:

11A25 Arithmetic functions; related numbers; inversion formulas
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