OEIS A000324: The Infinite Coprime Sequence and the Lucas Connection

In this Deep Dive, we investigate A000324 from the OEIS — a sequence born from a deceptively simple nonlinear recurrence that rockets from the start 1, 5, 9, 49, 2209 and beyond. What makes it truly fascinating is that it forms an infinite coprime sequence: any two distinct terms share no common factor greater than 1. This rare property isn’t just a curiosity; it hints at deep, hidden order behind rapid growth. We’ll place A000324 in a broader family of “exact mutual K-residue” sequences, specifically the case K = 4. This family includes famous members like Sylvester’s sequence (K = 1) and the Fermat numbers (K = 2), showing that a simple rule can nestle inside a rich hierarchy of number-theoretic objects. One of the most striking features is its connection to Lucas numbers. There are explicit expressions tying A000324 to Lucas numbers, revealing a direct and elegant bridge between a nonlinear recurrence and a classical, well-studied sequence. There’s also a remarkable identity: the infinite sum over n ≥ 1 of 4^n / A_n equals exactly 4, a kind of harmony that underscores the sequence’s internal structure. Beyond the formulas and identities, the story of A000324 is a reminder that simple rules can encode surprisingly intricate mathematics. It invites us to ask: what other seemingly tame OEIS sequences hide equally rich connections, and what broader patterns might they reveal about primes, residues, and the architecture of number theory?

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