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Interesting Esoterica Summation, volume 7

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Do you ever collect too much fun maths stuff to keep to yourself, and then start a website just so you’ve got somewhere to put it? That happens to me sometimes.

In case you’re new to this: every now and then I encounter a paper or a book or an article that grabs my interest but isn’t directly useful for anything. It might be about some niche sub-sub-subtopic I’ve never heard of, or it might talk about something old from a new angle, or it might just have a funny title. I put these things in my Interesting Esoterica collection on Mendeley. And then when I’ve gathered up enough, I collect them here.

In this post the titles are links to the original sources, and I try to add some interpretation or explanation of why I think each thing is interesting below the abstract.

Some things might not be freely available, or even available for a reasonable price. Sorry.

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Cushing your luck: properties of randomly chosen numbers

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Long-time Aperiodical muse David Cushing has made a bet with us that he can give us an interesting post every Friday for the next ten weeks. Every week that he sends a post, we buy him a bar of chocolate. Every week that he doesn’t send us a post, he buys us a bar of chocolate. For his first trick, David is going to do some unnatural things with the natural numbers.

The greatest common divisor (gcd) of two or more integers is the greatest integer that evenly divides those integers. For example, the gcd of $8$ and $12$ is $4$ (usually written as $\gcd(8,12)=4$). Two integers are called coprime (or “relatively prime”) if their gcd is equal to $1$.

A reasonable question to ask is,

Given two randomly chosen integers $a$ and $b$, what is the probability that $\gcd(a,b)=1$?

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What colour shirt do mathematicians wear?

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Star Trek uniforms

Readers of The Aperiodical may recall three excellent posts on the Maths of Star Trek by Jim ‘But Not As We Know It’ Grime. At the same time, Jim discussed the topic in glorious audio with Andy Holding and Will Thompson, hosts of the Science of Fiction podcast (worth listening to, but at least visit the page to see a picture of Jim nursing a tribble). As part of this, the hosts asked Jim what uniform colour mathematicians on the Enterprise would wear.

JIM: Science and medics, those are the blue shirts.

HOST: Where do mathematicians go? Scientists?

JIM: That’s right, yes, science.

HOST: You’re safe?

JIM: Yes, I am, I’m in the blue shirt category.

Jim is pleased to say that mathematicians wear blue because, as he explains, gold and red uniformed crew were much more likely to be killed during the famous five-year mission than those in blue. I’ve written in the past about maths and mathematicians being everywhere, for example when asserting that most of the Nobel prizes are for mathematics. Was Jim right about those blue-shirted mathematicians?

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Integer Sequence Review: A052486

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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.

Following last week’s palaver, we’re going to do our best to be serious this time. Game faces on.

A052486
Achilles numbers – powerful but imperfect: writing n=product(p_i^e_i) then none of the e_i=1 (i.e. powerful(1)) but the highest common factor of the e_i>1 is 1 (so not perfect powers).

72, 108, 200, 288, 392, 432, 500, 648, 675, 800, 864, 968, 972, 1125, 1152, 1323, 1352, 1372, 1568, 1800, 1944, 2000, 2312, 2592, 2700, 2888, 3087, 3200, 3267, 3456, 3528, 3872, 3888, 4000, 4232, 4500, 4563, 4608, 5000, 5292, 5324, 5400, 5408, 5488, ...

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