Take a Number

As I’ve mentioned before, I am fond of books, and have a hard time passing an outdoor bookseller’s table (and here in Gotham they are everywhere) without picking something up. As a result there are books all over my house; I simply don’t have enough bookshelves to contain them all, so they tend to accumulate in piles in less-trafficked areas. Every so often I make some attempt at reorganizing them, and the process takes much, much longer than it ought, because exhuming them from their dusty desuetude is like meeting old friends, and I wind up just sitting on the floor reading.

Today, while looking for something in the computer room upstairs, I noticed a forgotten pile of books, and second from the top was an old favorite: the Penguin Dictionary of Curious and Interesting Numbers, by David Wells (no, not the former Yankee hurler).

In this fascinating little volume, we learn about cyclic, amicable, untouchable and Kaprekar numbers, as well as aliquot sequences, the Fuerbach circle, and numbers sociable, pseudoprime, lucky, abundant and weird (a weird number, for example, is a number that is abundant without being the sum of any of its divisors; the smallest one is 70).

Opening the book at random, I find from the entry for 14 that it is the number of pounds in a stone, the number of days in a fortnight, and is the 3rd square pyramidal number: 14 = 1 + 4 + 9. Also (and I’ll bet you didn’t know this), 14 and 15 “are the first pair of successive numbers such that the sums of their factors, including the numbers themselves, are equal: 1 + 2 + 7 + 14 = 1 + 3 + 5 + 15 = 24.” There’s more, but let’s move on.

17, for example, is, among its other interesting properties, “equal to the sum of the digits of its cube, 4913. The only other such numbers are 1, 8, 18, 26, and 27, of which three are themselves cubes.

We learn that the seemingly unprepossessing number 153 is in fact equal to 1! + 2! + 3! + 4! + 5!, and that “when the cubes of the digits of any 3-digit number that is a multiple of 3 are added, and then this process is repeated, the final result is 153, where the process ends, because 153 = 13 + 53 + 33.” Also, we are reminded that in the New Testament Simon Peter drew a net from the sea of Tiberias that held 153 fishes.

Jumping ahead just a bit, it turns out that 4,679,307,774 is the only 10-digit number known that is equal to the sum of the 10th powers of its digits – a fact that, like me, you had probably forgotten.

But I think my favorite entry is the one for 39. Apparently 39 is, by all the relevant criteria, simply not interesting in any way. So what’s it doing in the list? Well, every number up to 39 was interesting enough in some way or other to make the cut, and 39 is the first one that wasn’t. It is, therefore, the smallest uninteresting number, which of course makes it interesting indeed. So how could they leave it out?

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