Hogben Numbers

Today I turned 25441 days and one of the properties of this number is that it's a Hogben number, the 160th such number in fact. I discovered this fact thanks to Numbers Aplenty in that site's description of the properties of this number. A link provided there contains the following information about these sorts of numbers:

The \(n\)-th Hogben number \(H_n\) is equal to \(n^2-n+1\).

Figure 1: SPIRAL ARRANGEMENT OF INTEGERS

In a spiral arrangement of the integers, Hogben numbers appear on the main diagonal (see Figure 1). Hogben numbers are often called central polygonal numbers. \(H_n\) is also the maximal number of ones that a \(n \times n\) {0,1} matrix can contain and still be invertible (that is an inverse matrix exists). The first Hogben numbers are 1, 3, 7, 13, 21, 31, 43, 57, 73, 91, 111, 133, 157, 183, 211, 241, 273, 307, 343, 381, ...

The Hogben numbers are listed in OEIS A002061 as the central polygonal numbers: \(a(n) = n^2 - n + 1\) with some interesting comments about where these Hogben numbers crop up. Let's consider some examples:

EXAMPLE 1

For n>1: a(n) is the maximum total number of queens that can coexist without attacking each other on an [n+1] x [n+1] chessboard. Specifically, this will be a lone queen of one colour placed in any position on the perimeter of the board, facing an opponent's "army" of size a(n)-1. 

The normal chess board consists of 8 x 8 = 64 squares, so the value of \(n^2-n+1\) when \(n=7\), namely 43, will give the maximum possible number of such queens. Figure 2 confirms this to be the case:

Figure 2: one against 42

EXAMPLE 2

Since \( (n+1)^2 - (n+1) + 1 = n^2 + n + 1 \) then from 7 onwards these are also exactly the numbers that are represented as 111 in all number bases: \(111_2=7 \), \(111_3=13 \), ... 

This is quite an interesting property and so we have the result that:

111 in base 2 is 7
111 in base 3 is 13
111 in base 4 is 21
111 in base 5 is 31
111 in base 6 is 43
111 in base 7 is 57
111 in base 8 is 73
111 in base 9 is 91
111 in base 10 is 111
111 in base 11 is 133
111 in base 12 is 157
111 in base 13 is 183
111 in base 14 is 211
111 in base 15 is 241
111 in base 16 is 273

The next example is taken from this source:

EXAMPLE 3

Figure 3: Hogben numbers within an isosceles triangle of numbers

This is a nice visual result and reveals an easy way to generate these numbers in a manner similar to Pascal's triangle.

So what about Lancelot Thomas Hogben after whom these numbers were named. He turns out to have been quite an interesting character. To quote from his Wikipedia entry:

Lancelot Thomas Hogben (9 December 1895 – 22 August 1975) was a British experimental zoologist and medical statistician. He developed the African clawed frog (Xenopus laevis) as a model organism for biological research in his early career, attacked the eugenics movement in the middle of his career, and popularised books on science, mathematics and language in his later career.

It is in his very popular 1936 book about Mathematics titled "Mathematics for the Million" that he presumably deals with numbers of the form \(n^2-n+1\). I have a copy of this book in electronic format in my library of ebooks but I couldn't find anything that sheds further light on these particular numbers. Here is a quote from the introduction to his book:

Figure 4: Lancelot Thomas Hogben 

Hogben was a conscientious objector in World War 1 and was imprisoned for a time. He was fiercely opposed to the Eugenics movement that was very active in the 1920s and 1930s. In World War 2, he was responsible for the British Army's medical statistics. He is quoted as saying:

"I like Scandinavians, skiing, swimming and socialists who realise it is our business to promote social progress by peaceful methods. I dislike football, economists, eugenicists, Fascists, Stalinists, and Scottish conservatives. I think that sex is necessary and bankers are not".

In addition to his extensive writing, he edited The "Loom of Language" by his friend Frederick Bodmer. This was a book that I borrowed from the library in Pontefract, Yorkshire, when I lived there in 1983-4, and which I read and very much enjoyed.