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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.
It is not clear whether or not "between" is to be consider to include either or both of those two numbers. In any case, the solution is not too different in each case. Let's assume that the perfect cube being sought is strictly smaller than the larger of the two numbers given. We take the cube root of that number and round it downward to the nearest integer and then cube it. If that number is greater than (or equal in case "between" is inclusive) to the smaller of the two numbers, then that is the perfect cube being sought. If it is smaller than the smaller of the two numbers, there is no such perfect cube.
There are two perfect numbers, 6 and 28, that are less than 100.
Prime numbers have two factors. The sum of their proper divisors is always 1.
There are infinitely many rational numbers between any two rational numbers. And the cardinality of irrational numbers between any two rational numbers is even greater.