Prime Generating Sequences

A couple of months ago (really, a long, long time ago) I found an interesting question on Mathematics Stack Exchange (another site to effectively waste away hours of your life). It reminded me of my Bachelor's thesis (which I wrote a really, really long time ago) about the sequence

where $a_1=7$ and

Here, $\mathrm{gcd}(a,b)=(a,b)$ stands for the greatest common divisor,1 i.e., the largest integer $d$ that divides both $a$ and $b$. This may not seem terribly interesting at first sight, but if you look at the first few values for $g_n$ you'll notice something curious:

1, 1, 1, 5, 3, 1, 1, 1, 1, 11, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 23, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 47, 3, 1, 5, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 101...

There are for sure a lot of ones in there, but other than that, all the numbers are primes. This is not a bias in the first few example -- Eric Rowland proved that all values of $g_n$ are either $1$ or a prime in a beautiful little paper back in 2008. Continue reading Prime Generating Sequences

1. It's common to abbreviate gcd(a,b)=(a,b) in number theory, and I shall do so in the remainder of the article. Similarly, it's convention to write lcm(a,b)=[a,b].