A "Perfect" Number (496 follower problem)

The number 496 is the third smallest perfect number: the sum of its proper divisors is the number itself: 1+2+4+8+16+31+62+124+248=496.1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248 = 496.

But it is also a "perfect" number in the sense that its third digit is the geometric mean of the first two digits: 49=6\sqrt{4\cdot 9} = 6.

How many three-digit integers abc\overline{abc} are there with the property, that cc is the geometric mean of aa and bb?


Note: Numbers starting in zero do not count. Thus, a "three-digit integer" lies between 100100 and 999999.

And here is a more challenging variation on the theme.

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