Perfect numbers

Question

Define a Perfect (capital-P) number as a natural number that is equal to the sum of its Divisors excluding 1 and the number itself. (So the Divisors of 28 are 2, 4, 7, 14, summing to 27.) Is there any such Perfect number? If yes, give an example. If no, prove it.

Answer 1

Here is a start. If there are no squared factors of the number $N=\prod p_i$, where prime factors are $p_i$ then the sum of the factors is $$\prod(p_i+1)-1-N=N$$ so that $\prod (1+p_i)$ is odd and $p_i$ is even. Clearly this is impossible for a solution.

Looking more closely we see that we get a product of terms of the form $$\prod \frac {p_i^{n_i+1}-1}{p_i-1}=2N+1$$where $N=\prod p_i^{n_i}$. Each term in the product has to be odd, and this means that $p_i=2$ or $n_i$ is even. And you need $2N+1$ to be suitably composite.

Answer 2

You're looking for a "Quasiperfect number". It's open whether such things exist (but if they do, they must be odd squares larger than $10^{35}$ and have at least seven prime factors).