140 and Other Numbers

There's this book which I borrowed called the Dictionary of Curious and Interesting Numbers by David Wells - and exactly seventy blogposts ago, I wrote one about 70 and other numbers. So I've decided to delve into the dictionary again, starting with 140...and I really should give the book back, it's been nearly six months.

Harmonic Mean/Numbers

140 is the smallest harmonic number - that is, the smallest number where its harmonic mean is an integer. But what is the harmonic mean? And what is a mean?

At school, everyone learns that the mean of a data set is the value of all the numbers in the set added together and divided by how many numbers there are in the set. It's the one used to calculate the standard deviation of a set. As in this formula:

 

That's the arithmetic mean, and it's the simplest by far and also the most used. But it's one of many - another is the geometric mean, which is the nth root of all the numbers in the set (say up to a number n) multiplied together:

And then you have the harmonic mean. This time, divide the reciprocals of all the numbers in the set by the number of numbers in the set:

                                                                    

For a given set, the arithmetic mean will be greater than the geometric mean, which will be greater than the harmonic mean, curiously enough. But that's not important. And there are other means too, like the root mean square, but I don't want to discuss means too much now.

140 is harmonic because the harmonic mean of its factors is an integer. As follows:

The factors of 140 are 1, 2, 4, 5, 7, 10, 14, 20, 28, 35, 70 and 140. That's twelve factors. And 12/(1/1 + 1/2 + 1/4 +...) is 5. Except for all the perfect numbers, 140 is the smallest harmonic number. 

e^π√163

It's called Ramanujan's constant, and is bafflingly close to being an integer. To the twelfth decimal place, in fact - just a string of .9999... beforehand.

It's what is known as an almost integer, and what's even more curious is that e^π√d can generate some more of these numbers. There's 19, 43 and 67. And like 163, they're somewhat part of the Heegner number group - if negative. The reason why these Heegner numbers can result in such impressive outcomes is due to some complex numbers and functions, and what's more curious is that -163 is as big (or as small?) as you can get with them. Heegner himself tried to prove this, but it wasn't complete - however, most agree he had a point. 

^{More on Heegner numbers here, because I would do a terrible joh explaining them any further}

Graham's Number (3↑↑↑...3)

The biggest number that the dictionary has an entry for is infamously described as being one that, if you memorised all its digits, it would break your brain. It's the upper bound for a problem whose answer is speculated to be just six, which is extremely funny because of just how massive Graham's Number actually is. The arrows are just powers; 3↑3 is 3³, for example. The total number of arrows in Graham's Number involves about sixty-four layers of four arrows, which is hard to compute in one's mind. 

Perhaps the ironic part about the number is that its size also means you can't just write it down. Even 3³³³, for instance, can't be written down entirely - we know its final digits, and that's it. Same with Graham's number - we have five-hundred final decimal places. So one of the biggest numbers we know is just a massive over-estimate. Here's a blogpost on the number which explains it better.

54

The Dictionary doesn't have an entry for 54 - it's the lowest integer without one. In a way, that makes it the least interesting integer...which also makes it quite interesting, as per the tongue-in-cheek proof by contradiction (more on that in this Wikipedia article).

But obviously 54 is more interesting than that fact. It's the sum of four consecutive square numbers (2 through to 5), and can also be represented as the sum of three squares in three different ways - and whilst two of those ways include repeats, it's still quite impressive. It reminds me of taxicab numbers - that is, numbers which are the sum of two cubes in two distinct ways. 54 is also a Leyland number, or one which is of the form a^b + b^a; 54 is 3³ + 3³.

54 does have a Wikipedia article, though. The same can't be said about 309, though, which is the smallest integer without an article over there. Or in other words, one can't accurately define a dull number.

Smith Numbers

The sum of a Smith number's digits is equal to the sum of its prime factors. One of the first numbers to be defined as such was 4937775, which was actually the titular Smith's phone number. The sum of its digits is 42, and the sum of the digits of its prime factors is also 42. I suppose this also means every prime is a Smith number, though maybe that's a bit too obvious.

The dictionary is certainly quite interesting; whilst from the outside it looks like lots of nerdy trivia with no real significance, that doesn't stop one from realising that sometimes the most arbitrary number will belong to some strange group with an unusual name, which only exists for fun. It goes far beyond what these blogposts attempt to do, though to be fair the author had an entire book to do that in. 

Maybe I'll write up "210 and Other Numbers"...eventually.

The aforementioned first blogpost: 70 and Other Numbers, featuring schizophrenic and perfect numbers.

You may also be a fan of my blogposts on i, e and φ, all linked here.