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Odd, spoof perfect factorizations. (English) Zbl 1484.11232

Let \(n\) be an integer. An expression of the form \(n = \prod_{i=1}^k x_i^{a_i},\) where \(x_i\) are integers and \(a_i\) are positive integers, is called a factorization of \(n\), and each \(x_i\) a base of the factorization. A factorization is odd, when \(n\) is odd; otherwise it is even. We define the following function to be evaluated on the collection of ordered pairs: \[\tilde{\sigma}(\{(x_i,a_i): 1 \leq i \leq k\}) = \prod_{i=1}^k \left(\sum_{j=1}^{a_i} x_i^j \right).\]
A factorization as above is called spoof perfect, if \[\tilde{\sigma}(\{(x_i,a_i): 1 \leq i \leq k\}) = 2n.\] A spoof perfect factorization \(\prod_{ i=1}^k x_i^{a_i}\) is said to be primitive, if for each proper subset \(S\) of \(\{1, 2, \ldots, k\}\), the factorization \(\prod_{i\in S} x_i^{a_i}\) is not spoof perfect. Furthermore, a spoof perfect factorization with a single base is called trivial.
In this paper, the spoof perfect factorizations are studied. More precisely, the trivial spoof perfect factorizations are characterized, and all nontrivial, odd, primitive spoof perfect factorizations with fewer than seven bases are computed. Moreover, it is proved that for each positive integer \(k\), there are finitely many nontrivial, odd, primitive spoof perfect factorizations with \(k\) bases. Finally, some interesting open questions are stated.

MSC:

11Y50 Computer solution of Diophantine equations
11D72 Diophantine equations in many variables
11A25 Arithmetic functions; related numbers; inversion formulas
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