Odd, spoof perfect factorizations (Q2071685)

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 \textit{a factorization} of \(n\), and each \(x_i\) a \textit{base} of the factorization. A factorization is \textit{odd,} when \(n\) is odd; otherwise it is \textit{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 \textit{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 \textit{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 \textit{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.