# Summer Excursion #3 — A Pi Containing Prime Number Sequence Challenge

Many of us know the first few digits of $$\pi$$ as $$3.14$$. Some of us know $$\pi$$ to higher accuracy — $$3.1415926$$. And the few elite know $$\pi$$ to thousands of digits. There are competitions for this!

Now, with probably zero practical math use, but 100% recreational math use, let’s see if we can find a sequence of prime numbers that contains $$\pi$$. Here’s how this works.

Let’s consider the increasing precision sequence for the digits of $$\pi$$ as $$p_{n}$$ with the first few terms $$3,31,314,3141,31415,314159,3141592,31415926,\ldots$$

The task is to find the sequence of smallest prime numbers that contains $$p_{n}$$ to various places. Thus, the smallest prime number that contains $$p_{0}$$ is $$3$$, which we will call $$a_{0}$$. From here, $$a_{1}$$ is the smallest prime number that contains $$p_{1}$$ — i.e., $$a_{1} = 31$$. And, $$a_{2} = 13147$$ since $$314$$ is in $$13147$$. We continue like this for all $$p_{n}$$.

The first several terms of $$a_{n}$$ are $$\{3,31,13147,73141,314159,314159,\ldots\}$$. Notice that $$314159$$ shows up twice since the smallest prime number that contains both $$31415$$ ($$p_{4}$$) and $$314159$$ ($$p_{5}$$) is $$314159$$.

The challenge is to find the next few terms! I don’t know of a good way to do this other than to program a search. Is this worthy of you, OEIS?? [Edit 7/25: Indeed! This sequence got published as A283247. Hurray!]

Good luck! 