Episode #125 of the Stack Overflow podcast is here. We talk Tilde Club and mechanical keyboards. Listen now

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.

289 questions
Filter by
Sorted by
Tagged with
231 views

Prove that if $2^a$ is superperfect, then $2^{a+1} - 1$ is a Mersenne prime.

Definition: Let $n \in \Bbb Z$ with $n>0$. Then $n$ is said to be superperfect if $σ(σ(n))=2n$. Where $σ$ is the sum of positive divisors arithmetic function. I am trying to solve a proof that ...
53 views

There are at most finitely many primes not dividing any elements in this sequence

A perfect number is a natural number $n$ that is the sum of its proper divisors. Otherwise said, they are the natural numbers $n$ such that $\sum_{d|n}d=2n.$ For instance, $6$ and $28$ are perfect. ...
278 views

Why are non-trivial powers called perfect powers?

Let number mean positive integer. What makes a number perfect is quite intricate: A perfect number is a number that is equal to the sum of its proper divisors. What makes a number a power is also ...
151 views

If $q^k n^2$ is an odd perfect number with Euler prime $q$, can $q=17$ hold?

Note: This question is an offshoot of this earlier MSE post. If $N$ is odd and $\sigma(N)=2N$ where $\sigma=\sigma_{1}$ is the classical sum-of-divisors function, then $N$ is said to be an odd ...
168 views

Explain these concepts of String Theory in easy words for mathematicians, from a popular point of view

In the Wikipedia's article for the number 496 in the section Physics is related the condition found by Green and Schwarz about this perfect number in their string theory. Question. Can you tell us ...
174 views

Can a multi-perfect number be a perfect square?

It is fairly easy to show that a perfect number $\Gamma$ cannot be written in the form $\Gamma=n^2$ for integer values of $n$. However, does this property hold true for multi-perfect numbers---that is,...
200 views

106 views

112 views

perfect powers and perfect numbers

The observation that 28 = 27 + 1 shows that it is possible to have consecutive perfect powers and perfect numbers . However, this must be very rare. Is it unique ? Questions: $\;$1) Are there any ...
126 views

58 views

Iterated $\sigma^k(n)$, excluding the possibility that primes $p|\sigma^t(n)$, $t< k$ divides an odd perfect number $n$

Let $m\geq 1$ an integer and $\sigma(m)=\sum_{d|m}$ the sum of positive divisors function. A positive integer is said to be perfect if and only if $\sigma(n)=2n$. We have that $\sigma(m)$ is a ...
99 views

Eulers Phi function with list of distinct perfect numbers

Prove that $\phi(n_1 n_2 \cdots n_m)$ = $2^{m-1}\cdot \phi(n_1)\cdot\phi(n_2) \cdots \phi(n_m)$ where all of the $n$'s are distinct even perfect numbers. I thought that because $\phi(2^k) = 2^{k-1}$ ...
68 views

Looking for one of B. Hornfeck's papers on odd perfect numbers

Let $\sigma(x)$ denote the sum of the divisors of $x$, and call the ratio $I(x) = \sigma(x)/x$ as the abundancy index of $x$. A number $N$ is called perfect if $\sigma(N)=2N$. According to this ...
On the binomial series $(1+\frac{1}{8n})^{1/2}$, where $n$ is an even perfect number
Since $\sqrt{1+8n}=\sqrt{8n}\sqrt{1+\frac{1}{8n}}$, and $\frac{1}{8n}<1$ when $n>1$ is an integer, then we can express the real number $\sqrt{1+\frac{1}{8n}}$ by its binomial series. This series ...