The Online Encyclopedia of Integer Sequences contains over 200,000 sequences. It contains classics, curios, thousands of derivatives entered purely for completeness’s sake, short sequences whose completion would be a huge mathematical achievement, and some entries which are just downright silly.

For a lark, David and I have decided to review some of the Encyclopedia’s sequences. We’ll be rating sequences on four axes: Novelty, Aesthetics, Explicability and Completeness.

A225143
Primes from merging of 10 successive digits in decimal expansion of $\zeta(2)$ or $\frac{\pi^2}{6}$.

9499012067, 4990120679, 3040043189, 1896233719, 2337190679, 9628724687, 2510068721, 8721400547, 9681155879, 5587948903, 7564558769, 9632356367, 3235636709, 3200805163, 4445184059, 3876314227, 2276587939, 1979084773, 9420451591, 9120818099, ...

Straight away, we can say that this sequence is finite. We’ll deduct a point at the end for that.

Novelty

David: It’s too contrived.

Christian: It’s both the decimal expansion of a famous constant, and a subset of the primes; both well-trodden boulevards of sequence cultivation. Taking just 10 digits shows a lack of imagination.

David: It’s stupid.

Christian: Agreed.

\[\frac{2}{5}\]

Aesthetics

Christian: Look at that! It’s basically random.

David: Sequences should increase. It’s ugly.

Christian: It doesn’t look like proper numbers. It looks like a code. If I was taking any number of digits, I’d want it to be odd.

David: It goes up and down too much. It’s ugly.

Christian: It would look nice if you wrote it in a grid.

David: At least it’s consistent.

\[\frac{2}{5}\]

Explicability

Christian: Pretty straight-forward. The primes are the primes; $\zeta(s) = \sum_{n=1}^{\infty} n^{-s}$ is the Riemann zeta function. $\zeta(2)$ is the sum of the reciprocals of the square numbers, which is $\frac{\pi^2}{6}$ thanks to Euler. Write it out in decimal, look for primes. Easy.

David: It’s straight-forward to understand, but what on Earth is the point in it? Why? I’d be happy to give it a 3.

Christian: I think it’s worth a 4.

David: Fine.

\[\frac{4}{5}\]

Completeness

Christian: It’s finite. So it can be completed. The OEIS currently only has the first 1,000 entries. Poor show.

David: Will it contain all the 10-digit primes?!

Christian: It can be completed but it isn’t. What do we think?

\[ \frac{3}{5} \text{(unsure)} \]

Final score

Deducting, as promised, one point for being loathsomely finite:

\[ \frac{2+2+4+3}{20} – \frac{1}{20} = \frac{10}{20} = \frac{1}{2} \]

Looks like this is a pretty average sequence.