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.

A051200
Except for initial term, primes of form “n 3’s followed by 1”.

3, 31, 331, 3331, 33331, 333331, 3333331, 33333331, 333333333333333331, 3333333333333333333333333333333333333331, 33333333333333333333333333333333333333333333333331, ...

Novelty

Christian: The decision to put a 3 at the start is certainly novel.

David: It’s quite common for people to look at primes following some sort of pattern when written in decimal, but this is unlike any I’ve seen. It’s pretty cool.

Christian: I don’t like it.

David: Why?

Christian: The 3 offends me.

David: That has nothing to do with its novyality (sic).

Christian: I can’t argue with that.

David: In fact, I quite like having a random 3 in front of things. For the rest of the article, can I be 3David?

Christian: If you want. Score?

\[ \frac{3}{5} \]

Aesthetics

Christian: I hate it. I hate it I hate it I hate it.

David: Apart from the first 3, it looks very nice.

Christian: Apart from her face, your mum looks very nice.

David: You disgust me.

Christian: I’m only following this sequence’s awful example.

David: It looks very nice at the start. It lulls us into a false sense of security that all “n 3s followed by a 1” will appear.

Christian: I’m not ready to move past the 3. WHY IS IT THERE? Shouldn’t the first term under consideration be 1, which isn’t prime?

David: That’s why it isn’t there.

Christian: SO WHY THIS USURPER THREE?

David: It’s followed by the empty 1.

Christian: I hate you.

David: Score?

Christian: I’m not going any higher than zero.

David: I’m not going any lower than three.

Christian: Let’s compromise.

\[ \begin{align}  f(t) &= \begin{cases} 0, & 0 \leq t \lt \frac{1}{2}, \\ 3, & \frac{1}{2} \lt t \leq 1. \end{cases} \\ t &\sim \mathcal{U}(0,1) \end{align} \]

Explicability

Christian: INEXPLICABLE.

David: An infinite number of terms are very easily explained.

Christian: Whyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyy?!?!?!?!?!?!?!?!?!?!?!?!?!?!?!?!?!?!

David: It would be rude to mark it down because of the first term.

Christian: There isn’t even a reference in the OEIS entry explaining why that first 3 is there. I am baffled. Bamboozled. Bastounded.

David: That isn’t even a word. And where’s the 3 from my name gone?

Christian: Whoops.

3David: That’s better. Score?

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

Completeness

3David: It’s so complete it’s got terms that shouldn’t even be in it. 5/5.

Christian: I give up. If you want to live in a world where sequences for no reason have a 3 in front of them, go ahead. 6/5.

\[  \frac{6}{5} \]

Final score

Since we defined a probabilistic score for aesthetics, we’ll just give the mean and variance of the total, $\tau$.

\[ \begin{align} \operatorname{E}(\tau) &= \frac{3+ \left( \frac{1}{2} \cdot 0 + \frac{1}{2} \cdot 3 \right) + 2 + 6}{20} = \frac{5}{8} \\ \operatorname{Var}(\tau) &= \frac{9}{800} \end{align} \]