OEIS A000289: Explosive nonlinear recurrence, infinite coprime property, and connections to Sylvester and Fermat sequences

Welcome, curious minds. In this episode we dive into A000289, the nonlinear recurrence that rockets from simple beginnings to monstrous numbers, while keeping every pair of terms coprime. We unpack the defining rule a(n) = a(n-1)^2 - 3·a(n-1) + 3 (and the related alternate forms), trace its tame early terms—1, 4, 7, 31—and explore its astonishing growth and the “infinite coprime” property. We’ll place it in the broader family of mutual‑k residues with k = 3, linking it to famous sequences like Sylvester’s and a Fermat‑related cousin. Along the way we’ll peek at different representations—doubly exponential ceiling forms and a product-based recurrence a(n) = 3 + ∏_{i< n} a(i)—showing how the same object can be viewed through multiple mathematical lenses. Join us as we uncover the hidden structure, surprising connections, and the beauty of simple rules leading to deep number-theoretic order.

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