Must Have A Lot Of Factors

Define a cool number as a positive integer \( n \geq 2 \) that satisfies the following:

For all pairs of factors \(a, b\) of \(n\) satisfying \(\gcd (a,b)=1\), then \(a+b-1\) is also a factor of \(n\).

Find the number of cool numbers under 1000.

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