Prime Squared Semi-Magic Squares

I've made numerous posts about magic squares of one form or another and even posted about magic cubes and magic polygons. On 17th May 2021, I posted about Prime Semi-Magic Squares and today's post is also about semi-magic squares and primes, except that in this post I'll be looking at the squares of primes. Remember that a semi-magic square only needs to have its rows and columns adding to a constant, not its diagonals.

The number associated with my diurnal age today is 27531 and it has a property that qualifies it for membership is OEIS A269344:

A269344

Magic sums of 3 X 3 semimagic squares composed of squares of primes.

The initial members of this sequence are:

5691, 26859, 27531, 85659, 93219, 111699, 113331, 155739, 179091, 203571, 228459, 239379, 277419, 281499, 344931, 376971, 469011, 487491, 490299, 520491, 583779, 631491, 679539, 806547, 851211, 896091, 922659, 1004379, 1067619, 1099539, 1119459, 1134771

The magic square associated with 27531, the third member of the sequence, is shown in Figure 1. All the rows and columns add to this number.

Figure 1: source

Now 27531 can be expressed as a sum of three primes squared in 16 different ways. See Figure 2.

Figure 2: permalink

Looking at these 16 different combinations of primes, it's easy to see the ones that were selected to form the magic square. Let's look at the second number in the sequence, 26859. See Figure 3 where all rows and columns add to 26859.

Figure 3: source

Now 26859 can be expressed as a sum of three primes squared in 18 different ways. See Figure 4.

Figure 4: permalink

This site shows the magic squares for all members of the sequence up to 1099539.