Questions tagged [perfect-numbers]

Questions about or involving perfect numbers which are positive integers that are equal to the sum of their proper positive divisors.

273 questions
102 views

Show that if an odd perfect number exists, it must be divisible by at least 3 different primes

I would assume you'd start by showing that $\dfrac{p}{p-1}\cdot\dfrac{q}{q-1}< 2$ but I don't know how to show that nor how to continue afterwards
269 views

Are all known $k$-multiperfect numbers (for $k > 2$) *not* squarefree?

A positive integer $N$ is said to be $k$-multiperfect if $$\sigma(N) = kN$$ where $\sigma(x)$ is the sum of the divisors of $x$ and $k$ is a positive integer. (The case $k = 2$ reduces to the ...
3k views

Can powers of primes be perfect numbers?

I need to prove the following, though I'm not 100% certain I understand the definition of a perfect number. Prove that no perfect number is a power of a prime. First of all, I'm assuming that the ...
335 views

Can an odd perfect number be divisible by $101$?

Preamble - This question is an offshoot from the following earlier questions here at MSE: Can an odd perfect number be divisible by 825? Can an odd perfect number be divisible by 165? Odd perfect ...
1k views

If $n$ is an even perfect number $n> 6$ show that the sum of its digits is $\equiv 1 (\bmod 9)$

If $n$ is an even perfect number $n> 6$ show that the sum of its digits is $\equiv 1 \mod 9$. I know perfect numbers are of the form $(2^{p-1})(2^{p}-1)$. I have a few trials that I have done and ...
632 views

Can an odd perfect number be divisible by $165$?

I know that an odd perfect number cannot be divisible by $105$ or $825$. I wonder if that's also the case for $165$ (this is actually a stronger statement).
881 views

Can an odd perfect number be divisible by $825$?

I know that an odd perfect number cannot be divisible by $105$. I wonder if that's also the case for $825$.
243 views

How are 10-20 digit multiperfect and hemiperfect numbers efficiently computed?

This numericana item on multiperfect and hemiperfect numbers contains some impressively enormous numbers. How were these actually computed ? The associated OEIS pages (A007691 & A159907) just ...
378 views

Prove that if n is an even perfect number then $\sigma(\sigma(n)) < 6n$

$\sigma(n)$ refers to the sum of all divisors function. If n is an even perfect number, then $\sigma(n) = 2n$, but why is $\sigma(\sigma(n)) < 6n$?