The square of any odd number will be an odd square number.
eg 12 = 1
32 = 9
52 = 25
etc
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There are infinitely many of them. The square of every odd number will be an odd square number.
Assuming you know that your number is a perfect square, the square root of an even number is even, and the square root of an odd number is odd.
A square number has an odd number of factors, but a number with an odd number of factor pairs is nothing special.
A square number
The square of an even number is always even.