### Exactly how bad is the 13 times table?

Let’s recite the $13$ times table. Pay attention to the first digit of each number:

\begin{array}{l} \color{blue}13, \\ \color{blue}26, \\ \color{blue}39, \\ \color{blue}52 \end{array}

What happened to $\color{blue}4$‽

A while ago I was working through the $13$ times table for some boring reason, and I was in the kind of mood to find it really quite vexing that the first digits don’t go $1,2,3,4$. Furthermore, $400 \div 13 \approx 31$, so it takes a long time before you see a 4 at all, and that seemed really unfair.

### The OEIS now contains 300,000 integer sequences

The Online Encyclopedia of Integer Sequences just keeps on growing: at the end of last month it added its 300,000th entry.

Especially round entry numbers are set aside for particularly nice sequences to mark the passing of major milestones in the encyclopedia’s size; this time, we have four nice sequences starting at A300000. These were sequences that were originally submitted with indexes in the high 200,000s but were bumped up to get the attention associated with passing this milestone.

### πkm Running Challenge: 7-day update

This month I'm doing a completely irrational sponsored run for Sport Relief, aiming to raise £100π by running πkm per day, every day in March. I'm one week in, and here's the story so far.

### Carnival of Mathematics 155

The next issue of the Carnival of Mathematics, rounding up blog posts from the month of February, and compiled by Ben, is now online at Math Off The Grid.

The Carnival rounds up maths blog posts from all over the internet, including some from our own Aperiodical. See our Carnival of Mathematics page for more information.

### Prime Time

We spotted this photograph of a letter to The Telegraph, shared by Card Colm on Twitter earlier in the year. It’s exactly the kind of mathematical claim we like to enjoy verifying, so we thought we’d dig in.

### I’m going to run πkm every day in March

Inspired by the BBC’s Sport Relief fundraising campaign, I’ve decided to set myself a vaguely mathematical running challenge. My current routine does involve a little running, but nothing serious, so I’ve given myself a bar to aim for that’s both vaguely achievable, and completely irrational.

I’ll aim to run π kilometres (or as close as I can get, with the measuring instruments I have access to) each day during the month of March. This will either be on the treadmill at my gym – in which case I’ll try to get a photo of the ‘total distance’ readout once I’ve finished – or out in the real world, for which I’ll use some kind of running GPS logging device, to provide proof I’ve done it each day. Some days I’ll run on my own, and others I’ll be accompanied by friends/relatives, who’ll be either running as well or just making supportive noises. At the end of the month, I’ll post an update documenting my progress/success/failure.

Serious request: if you know of anywhere in the UK I can reasonably get to where there’s an established circle that’s exactly 1km in diameter, I can try to come and run round the circumference of it. Drop me an email if so.

If you’d like to support my ridiculous plan, you can follow my progress and donate on my fundraising page, or encourage others to do so by visiting pikm.run (I paid £4 for the URL, so now I have to do it). Sport Relief is the even-numbered-years-counterpart of Comic Relief, which together raise money for thousands of projects all over the UK and in the developing world, to help the vulnerable and those in need.

### Taming the AGM

This post is in response to Peter’s post introducing the Approximate Geometric Mean.

The approximate geometric mean $\mathrm{(AGM)}$ is a nice approximation of the geometric mean $\mathrm{(GM)}$, but it has some quirks as we will see. After a discussion at the MathsJam gathering, I was intrigued to find out how good an approximation it is.

To get a better understanding, we first have to look again at its definition. For $A=a\cdot 10^x$ and $B=b \cdot 10^y$, we set

$\mathrm{AGM}(A,B):=\mathrm{AM}(a,b)\cdot 10^{\mathrm{AM}(x,y)}$

where $\mathrm{AM}$ stands for the arithmetic mean. This makes also sense when $a$ and $b$ are not just integers between 1 and 10, but any real numbers. Note that we won’t consider negative $A$ and $B$ (i.e. negative $a$ and $b$), as the geometric mean runs into issues if we do so. The values of $x$ and $y$ may be negative, though. The $\mathrm{AGM}$ looks like a mix between the $\mathrm{AM}$ and the $\mathrm{GM}$, so what can possibly go wrong?