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Notes on a conjecture for Singmaster. (English) Zbl 0845.11006

A sequence \(\{a_i\}^n_{i= 1}\) of positive integers in nondecreasing order is a sum=product sequence of size \(n\) if \(\sum^n_{i= 1} a_i= \prod^n_{i= 1} a_i\). It is easily shown that \(\{2, 2\}\), \(\{1, 2, 3\}\), and \(\{1, 1, 2, 4\}\) are the only sum=product sequences having sizes 2, 3, and 4 respectively. Let \(N(n)\) denote the number of different sum=product sequences of size \(n\). The authors give an explicit expression for \(N(n)\). Also, let \(N(n, k)\) denote the number of different (sum)\(^k\)=product sequences of size \(n\) \((n> k\geq 2)\). Then it is proved that \(N(n, k)= \infty\).
Reviewer: E.L.Cohen (Ottawa)

MSC:

11A25 Arithmetic functions; related numbers; inversion formulas
11D72 Diophantine equations in many variables
11D41 Higher degree equations; Fermat’s equation
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