text: For any natural number n, there exists only one of these numbers that can be expressed in the form "n-cubed plus 1".
full_question: For any natural number n, there exists only one of these numbers that can be expressed in the form "n-cubed plus 1". Kanold was the first to show that the amount of these numbers below a given integer n had an asymptotic form of little-O of the square root of n. With the exception of the smallest of these, all known so far can be written as the sum of the cubes of consecutive positive odd integers. For a Mersenne prime with exponent p, a number of this type can be found by multiplying the Mersenne prime by 2 to the power p minus 1, according to the Euler-Euclid conjecture. These numbers are a subset of the triangular numbers, and all numbers of this type found so far are even. For 10 points, name these numbers, such as 496 and 6, that are equal to the sum of their proper divisors.
first_sentence: For any natural number n, there exists only one of these numbers that can be expressed in the form "n-cubed plus 1".
raw_answer: perfect numbers