That Number Again

The number associated with my diurnal age today, 27079, is a member of OEIS  A133562:

A133562

Numbers which are the sum of the squares of seven consecutive primes.

In the case of 27079, the primes are as shown below (permalink):$$47^2+ 53^2+ 59^2+ 61^2+ 67^2 +71^2+ 73^2 = 27079$$However, it is the very first term in this sequence that is most interesting as it is the number 666:$$2^2+3^2+5^2+7^2+11^2+13^2+17^2=666$$It is the only even number in the sequence because it includes the even square \(4=2^2\). I naively thought that it might be possible to arrange these squares to form an \(18 \times 37\) rectangle but I was swiftly disabused of this notion when I recalled a post called Squaring the Square that I'd made on January 29th 2021.

In this post, I note that the smallest square that can be constructed of smaller squares of unequal size requires 21 squares and has a side of 112 units. The smallest rectangle than can be constructed from smaller squares of unequal size requires 9 squares and has dimensions of \(32 \times 33 \) units. Clearly then a rectangle of dimensions \(18 \times 37\) cannot be constructed from only seven unequal squares.

However, I did find a decomposition of the square of side 666 into 26 smaller squares of unequal size. These squares have the following sides: 2, 3, 8, 33, 36, 55, 65, 69, 89, 90, 97, 102, 105, 107, 109, 111, 120, 129, 132, 171, 175, 185, 186, 220, 230, 261. See Figure 1.

Figure 1: decomposition of square of side 666 (source)

This square of side 666 units has an area of 443556 square units. So just an interesting little diversion with the result that:$$2^2+ 3^2 + \dots + 230^2+ 261^2 = 666^2$$