A New Primes-Generating Sequence

Abstract

For the sequence defined by \[ a(n) = \frac{n^2 - n - 1}{\gcd\big(n^2 - n - 1,\, b(n-3) + n\,b(n-4)\big)} \] Where $b(n)=(n+2)(b(n-1)-b(n-2)),$ with initial conditions $b(-1)=0$ and $b(0)=1$, we find that $a(n)$ contains only $1$'s and primes, and can be represented as a finite continued fraction. It is more efficient for generating prime numbers than the Rowland sequence.

5 pages