*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’re rating sequences on four axes: Novelty, Aesthetics, Explicability and Completeness.*

This is the triumphant return of the integer sequence reviews!

A193430

Primes p such that p+1 is in A055462.23, 6911, 5944066965503999, ...

**Christian:** Superduperfactorial primes! Primes which are one more than a superduperfactorial. Which are super duper.

**David:** I don’t believe there are infinitely many of them. How do you prove that?

**Christian:** Umm..

**David:** It says “the next term, if it exists”! So there isn’t a next term. Or there might be.

**Christian:** Shall we just go through the categories.

**David:** Right. What’s the first one? Explicability?

### Explicability

**David:** What’s a superduperfactorial? And more importantly, what’s the next one going to be called? I vote for superduperlooperfactorial. It’s what Santa does in Santa Claus The Movie. He catches Patch.

**Christian:** Patch?

**David:** I’ve never understood that bit. Apparently the only way to catch Patch is by doing a loop.

**Christian:** Maybe he can change his trajectory’s radius of curvature but not his velocity?

**David:** Santa can teleport as well. So I don’t understand why he’s using the sleigh at that point in the movie.

**Christian:** We’re no closer to understanding what a superduperfactorial is.

**David:** Ah! Superduperlooperfactorials are in the OEIS (A057527), but they’re called “fourth level factorials”.

**Christian:** I prefer your coinage. Let’s make an edit. Anyway, WHAT IS A SUPERDUPERFACTORIAL DAVID?

**David:** The $n$^{th} superduperfactorial is the product of the first $n$ superfactorials.

**Christian:** And a superfactorial is…

**David:** The product of the first $n$ factorials.

**Christian:** … which is the product of the first $n$ natural numbers. (If $0$ is unnatural, which it isn’t)

**David:** Ooh, I wonder if there’s a version of $\Gamma$ for superduperfactorials.

**Christian:** Well, one exists, but how’s it defined?

**David:** Did you know that $\Gamma$ is the only locally convex continuous function which agrees with the factorials?

**Christian:** I suppose I’d always assumed it, but it’s good to know.

**David:** The derivation is bananas. You take $\sin$ and put your hand over half the zeroes, then you take $e$ to the $\gamma$, where $\gamma$ is the Euler-Mascheroni constant, and times that by something so $\Gamma(1)=1$, then pull a bunny out of a hat.

**Christian:** I will never understand analysis. And I also move to give this $\frac{1}{5}$ for Explicability.

**David:** We haven’t even described what the sequence is yet, Christian. It’s about primes.

**Christian:** OK, this sequence is the primes that are one less than a superduperfactorial number.

**David:** I’m surprised that $n!-1$ won’t always factorise. Because $n-1$ usually factorises.

**Christian:** That’s among the worst things I’ve heard today.

**David:** Yeah, well $x^2-1$ always factorises. $2^n-1$ always factorises, if $n$ isn’t prime.

**Christian:** Yeesh. We had a nice explanation a few lines up, and now you’re conjecturing like a loon again. Explicability score? I reckon 4.

**David:** Well it is very explicable, but I think we’ve done such a bad job of explaining it that it only deserves 1 or 2.

**Christian:** Let’s be generous.

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

### Novelty

**Christian:** Well, apart from the obvious prior art in Disney’s *Mary Poppins*, I think this is new to me.

**David:** I think it’s pretty novel. Well, I think the elements are pretty novel.

**Christian:** In that you haven’t seen $5944066965503999$ in the wild before?

**David:** I’m just googling the OEIS to see if it appears in another sequence.

**Christian:** “googling the OEIS”.

**David:** It only appears in the one that this one is a subset of (A238265) I think it appears in A000027 too, but they don’t give enough terms to be sure.

**Christian:** It definitely is. But that shouldn’t affect this sequence’s Novelty score. We’ve been down that crazy-paved road before.

**David:** I think it’s novel but not fully five-points novel because we’ve already seen similar sequences with a few primes which grow very very fastly and may or may not be infinite.

**Christian:** Agreed, apart from “fastly”.

**David:** Good word.

**Christian:** Bad David. $\frac{3}{5}$?

**David:** Uhh, yep.

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

### Aesthetics

**Christian:** I quite like the fourth term.

**David:** There are lots of double digits. It’s a pretty number. And 23 is a good number, because that’s the number Michael Jordan used to play for the Chicago Bulls. I’ve been watching a lot of basketball history documentaries recently. That’s what TV is in America, apparently.

**Christian:** Truly a great cultural gift to the world. I also like the name of this sequence, or really, the name of the things this is one less than.

**David:** You typed my point while I was saying it.

**Christian:** So do the comely looks of the superduperfactorials rub off on this one? As usual, we’re reviewing the wrong sequence.

**David:** We’ve been doing this for ages. Surely we know better.

**Christian:** THE FIRST REVIEW WAS ON MAY THE FOURTH!

**David:** That’s pretty cool!

**Christian:** I feel benevolent. $\frac{5}{5}$?

**David:** Yep.

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

### Completeness

**Christian:** No. None. We have four terms, don’t know if there are any more, and –

**David:** What did we give the Wieferich primes? Because the way we normally do this is, we look at a similar sequence and award the opposite of whatever we gave that.

**Christian:** Let’s double-triple-quadruple guess ourselves and make up a number.

\[ \frac{3.143}{5} \]

**David:** Bet you thought we were going for $\pi$ there!

**Christian:** Loser!

### Total score

**David:** Aww, are we done already? I was just starting to have fun.

**Christian:** Sadly yes, but we’ve fitted quite marvellously in the time I had allotted before pre-MathsJam Micky D’s.

**Christian:** And now, the scores:

\[ \frac{2+3+5+3.143}{20} = \frac{13.143}{20} = \frac{13143}{20000} \]

**Christian:** Well done picking a number which is coprime with everything, David.

**David:** If you write $13143$ in base $5$ and times by $2$ and write that in base $5$, then they’re anagrams of each other.

*<Christian whimpers>*

**David:** That is sequence A023061.

**Christian:** I have no words.

**David:** I think that’s the true sequence of the day. And I think we should award it the March 2014 Integer Sequence Royal Celebrity Knockout champion. I trust you’ll have the trophy ready when I come back.

**Christian:** Your trust is well-placed.

**David:** Sometimes I read these back and I think they’re the stupidest thing in the world.

**Christian:** That’s a good thought to end on.

It’s a bit fiddlier than that: $\Gamma$ is the unique locally convex function satisfying $\Gamma(x+1) = x \Gamma x$ and $\Gamma (1) = 1$.

You could make a convex interpolation of factorial by taking its lower convex hull, and then there are ways to modify that to get more smoothness if you really want it.