A Plethora Of Squares

The number associated with my diurnal age today, 27625, has the unique quality that it is the smallest number capable of being expressed as a sum of two squares in exactly eight different ways. Here are the different ways:$$\begin{array}{rl}{20}^2+{165}^2& =27625\\ {27}^2+{164}^2& =27625\\ {45}^2+{160}^2& =27625\\ {60}^2+{155}^2& =27625\\ {83}^2+{144}^2& =27625\\ {88}^2+{141}^2& =27625\\ {101}^2+{132}^2& =27625\\ {115}^2+{120}^2& =27625\end{array}$$The number arises from 27625's factorisation where:$$27625=5^3\times 13\times 17$$To determine the number of ways in which it can be written as the sum of two squares, we add 1 to each index, multiply them together and divide the product by 2. If the product is not even, then we round the result up. In the case of 27625 we have:$$\begin{array}{rl}\frac{(3+1)\times (1+1)\times (1+1)}{2}& =\frac{4\times 2\times 2}{2}\\ & =8\end{array}$$This property of 27625 qualifies it for membership in OEIS A016032:

 A016032: least positive integer that is the sum of two squares of positive integers in exactly $n$ ways.

The initial members of the sequence are:

2, 50, 325, 1105, 8125, 5525, 105625, 27625, 71825, 138125, 5281250, 160225, 1221025, 2442050, 1795625, 801125, 446265625, 2082925, 41259765625, 4005625, 44890625, 30525625, 61051250, 5928325, 303460625, 53955078125, 35409725, 100140625, 1289367675781250

It can be noted that the sequence is not monotonic increasing. For example, 8125 is the smallest number that can be expressed as a sum of two squares in exactly five ways but 5525 is the smallest number that can be expressed as a sum of two squares in exactly six ways.

Now if we square 27625 we have the following factorisation:$${27625}^2=5^6\times {13}^2\times {17}^2$$Now this number can be expressed as a sum of two squares in 32 different ways. These are the possible ways:$$\begin{array}{rl}{0}^2+{27625}^2& ={27625}^2\\ {969}^2+{27608}^2& ={27625}^2\\ {1175}^2+{27600}^2& ={27625}^2\\ {2625}^2+{27500}^2& ={27625}^2\\ {3060}^2+{27455}^2& ={27625}^2\\ {3588}^2+{27391}^2& ={27625}^2\\ {4225}^2+{27300}^2& ={27625}^2\\ {5180}^2+{27135}^2& ={27625}^2\\ {5655}^2+{27040}^2& ={27625}^2\\ {6600}^2+{26825}^2& ={27625}^2\\ {6800}^2+{26775}^2& ={27625}^2\\ {7223}^2+{26664}^2& ={27625}^2\\ {7735}^2+{26520}^2& ={27625}^2\\ {8856}^2+{26167}^2& ={27625}^2\\ {9724}^2+{25857}^2& ={27625}^2\\ {10220}^2+{25665}^2& ={27625}^2\\ {10625}^2+{25500}^2& ={27625}^2\\ {10815}^2+{25420}^2& ={27625}^2\\ {11700}^2+{25025}^2& ={27625}^2\\ {12137}^2+{24816}^2& ={27625}^2\\ {13000}^2+{24375}^2& ={27625}^2\\ {13847}^2+{23904}^2& ={27625}^2\\ {14025}^2+{23800}^2& ={27625}^2\\ {14400}^2+{23575}^2& ={27625}^2\\ {15620}^2+{22785}^2& ={27625}^2\\ {16575}^2+{22100}^2& ={27625}^2\\ {17340}^2+{21505}^2& ={27625}^2\\ {17500}^2+{21375}^2& ={27625}^2\\ {18239}^2+{20748}^2& ={27625}^2\\ {18600}^2+{20425}^2& ={27625}^2\\ {18921}^2+{20128}^2& ={27625}^2\\ {19305}^2+{19760}^2& ={27625}^2\end{array}$$Once again, we know that there are 32 ways to write this number as a sum of two squares because looking at the indices again we have:$$\begin{array}{rl}\frac{(6+1)\times (2+1)\times (2+1)}{2}& =\frac{7\times 3\times 3}{2}\\ & =\frac{63}{2}\\ & \to 32\text{ rounded up}\end{array}$$Now this property of the square of 27625 qualifies it for membership in OEIS A097244:

A097244: numbers $n$ that are the hypotenuse of exactly 31 distinct integer-sided right triangles, i.e., $n^2$ can be written as a sum of two squares in 31 ways.

By 31 ways and not 32 ways is meant that the number can be written a sum of two distinct non-zero numbers in 31 ways. The initial members of this sequence are:

27625, 47125, 55250, 60125, 61625, 66625, 78625, 82875, 86125, 87125, 94250, 99125, 110500, 112625, 118625, 120250, 123250, 129625, 133250, 134125, 141375, 144625, 148625, 155125, 157250, 157625, 164125, 165750, 172250, 174250, 177125

As can be seen, 27625 is the first member of this sequence.