Anti-Magic Squares

Today I turned 25307 days old and, as is my habit, I examined the entries for this number in the Online Encyclopaedia of Integer Sequences or OEIS. I've already mentioned 25307 in my previous post as forming part of a prime quadruple consisting of 25301, 25303, 25307 and 25309. For this reason, it is mentioned in OEIS A136721 Prime quadruples: 3rd term.

However, I also came across a mention of the number in OEIS A117560 \(n(n^2-1)/2 - 1\) which at first sight seemed unremarkable. However, on reading further I discovered that the numbers in this sequence form an:

approximation for the lower bound of the "anti-magic constant" of an anti-magic square of order n. The anti-magic constant here is defined as the least integer in the set of consecutive integers to which the rows, columns and diagonals of the square sum. 

I was of course familiar with magic squares but I'd never heard of anti-magic squares. I was prompted to investigate further.

To quote from WolframAlpha:

An anti-magic square is an \(n×n\) array of integers from \(1\) to \(n^2\) such that each row, column, and main diagonal produces a different sum such that these sums form a sequence of consecutive integers. It is therefore a special case of a heterosquare. It was defined by Lindon (1962) and appeared in Madachy's collection of puzzles (Madachy 1979, p. 103), originally published in 1966.

There are no \(2 \times 2\) or \(3 \times 3\) anti-magic squares. The first occurrence is of a \(4 \times 4 \) anti-magic square and shown below are examples of anti-magic squares from \(4 \times 4\) to \(9 \times 9\):

Source: http://www.magic-squares.net/anti-ms.htm

The formula \(0.5 \times n(n^2-1)-1 \) yields 251 in the case of \(n=8 \) but it should be remembered that this number is a lower bound. There are \( 8 \times 8\) anti-magic squares that start with a higher number. Below is a diagram of an \( 8 \times 8\) anti-magic square showing the various totals from 252 to 269 (as opposed to 251 to 268 in the example above):

In the case of 25307, the value of \(n\) is 37. So the sum of any sequences formed by adding the elements in any row, column or diagonal of this \(37 \times 37\) anti-magic square of numbers have 25307 as their lower bound.

There is a similar sequence for the upper bounds of anti-magic squares. The formula for generating the terms is:$$ \lfloor \frac{n(n^3-n-3)}{2(n-1)} \rfloor $$The OEIS is A117561.