Catalacci Sequence

Introduction

This paper introduces a new integer sequence called the Catalacci sequence — a name blending "catalyst" and "Fibonacci" — which modifies the classic Fibonacci recurrence with a prime-number trigger. Starting from C(0) = C(1) = 1, each subsequent term is built as C(n) = C(n-1) + C(n-2) + P(n), where P(n) equals 1 if the sum of the two previous terms happens to be a prime number, and 0 otherwise. So it behaves exactly like Fibonacci, except every time the running sum lands on a prime, the sequence gets an extra "+1 boost." The central result: despite this irregular, prime-dependent perturbation, the paper proves that the ratio of consecutive terms still converges to the golden ratio φ ≈ 1.618033988749894 — exactly the same limiting ratio as ordinary Fibonacci. The argument is that prime boosts occur increasingly rarely as the sequence grows (roughly logarithmically often, per the Prime Number Theorem), so their cumulative effect becomes vanishingly small compared to the sequence's exponential growth. This is backed by a formal convergence proof, a proof that the perturbation error shrinks at a rate of O(1/n), and computational verification up to n = 1,000,000 terms, where the ratio matches φ to over 19 decimal places. The paper generalises the idea into a "Catalacci family" — a Twin Prime variant (boost triggered by twin primes) and a Sophie Germain variant (boost triggered by Sophie Germain primes, significant in cryptography), both conjectured, with computational support, to also converge to φ. Proposed applications include pseudorandom number generation, cryptographic key derivation, stream ciphers, and use as an educational tool for visualising prime distribution. It closes with open problems (rigorously proving the twin-prime and Sophie-Germain variants converge, characterising the boost distribution, higher-order generalisations), a Python implementation using Miller–Rabin primality testing, and a draft OEIS submission for the sequence.