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No, there are no known perfect numbers between 1 and 30. The only perfect numbers that have been discovered are 6, 28, 496, and 8128.
The square numbers between 1 and 20 are 1,4,9 and 16.
All whole numbers between 1 and 400 (=20 square).
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Perfect numbers have the form 2n-1(2n-1) where 2n-1 is a Mersenne prime. When a new Mersenne prime is discovered, so is a new perfect number.