Conway's subprime Fibonacci sequences

TL;DR

This paper introduces a novel variation of Fibonacci sequences called 'subprime' sequences, where each term is derived by summing the last two and dividing by the smallest prime divisor if the sum is composite, revealing pseudo-random behavior and cyclical termination.

Contribution

It systematically studies the elementary properties and behaviors of 'subprime' Fibonacci sequences, a new class of sequences with unique pseudo-random and cyclical characteristics.

Findings

Sequences exhibit pseudo-random behavior.

Sequences generally terminate in a few cycles.

Properties are similar to 3x+1 sequences.

Abstract

It's the age-old recurrence with a twist: sum the last two terms and if the result is composite, divide by its smallest prime divisor to get the next term (e.g., 0, 1, 1, 2, 3, 5, 4, 3, 7, ...). These sequences exhibit pseudo-random behaviour and generally terminate in a handful of cycles, properties reminiscent of 3x+1 and related sequences. We examine the elementary properties of these 'subprime' Fibonacci sequences.