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:$$ \begin{align} 20^2+ 165^2 &= 27625\\27^2+ 164^2 &= 27625\\45^2+ 160^2 &=27625\\60^2+ 155^2 &=27625\\83^2+ 144^2 &=27625\\88^2+ 141^2 &= 27625\\101^2+ 132^2 &=27625\\115^2+ 120^2 &=27625 \end{align}$$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:$$ \begin{align} \frac{(3 +1) \times (1 +1) \times (1+1)}{2} &= \frac{4 \times 2 \times 2}{2} \\ &=8 \end{align}$$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:$$ \begin{align} 0^2+ 27625^2 &= 27625^2\\969^2+ 27608^2 &=27625^2\\1175^2+ 27600^2 &=27625^2\\2625^2+ 27500^2 &=27625^2\\3060^2 +27455^2 &=27625^2\\ 3588^2+ 27391^2 &=27625^2\\ 4225^2+27300^2 &=27625^2\\5180^2+ 27135^2 &=27625^2\\5655^2+ 27040^2 &=27625^2\\6600^2+ 26825^2 &=27625^2\\6800^2+ 26775^2 &=27625^2\\7223^2+ 26664^2 &=27625^2\\7735^2+ 26520^2 &=27625^2\\8856^2+ 26167^2 &=27625^2\\9724^2+ 25857^2 &=27625^2\\10220^2+ 25665^2 &=27625^2\\10625^2+ 25500^2 &=27625^2\\10815^2+ 25420^2 &=27625^2\\11700^2+ 25025^2 &=27625^2\\12137^2+ 24816^2 &=27625^2\\13000^2+ 24375^2 &=27625^2\\13847^2+ 23904^2 &=27625^2\\14025^2+ 23800^2 &=27625^2\\14400^2+ 23575^2 &=27625^2\\15620^2 +22785^2 &=27625^2\\16575^2+ 22100^2 &=27625^2\\17340^2+ 21505^2 &=27625^2\\17500^2+ 21375^2 &=27625^2\\18239^2+ 20748^2 &=27625^2\\18600^2+ 20425^2 &=27625^2\\18921^2+ 20128^2 &=27625^2\\19305^2+ 19760^2 &=27625^2 \end{align}$$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:$$\begin{align} \frac{(6+1) \times (2+1) \times (2+1)}{2} &= \frac{7 \times 3 \times 3}{2}\\ &= \frac{63}{2}\\ &\rightarrow 32 \text{ rounded up} \end{align}$$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.