# Questions tagged [perfect-numbers]

A perfect number is a positive integer that is equal to the sum of its proper positive divisors.

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### Does this Theorem 2 from Dandapat et al. imply that $\gcd(\sigma(p^k),\sigma(a^2)) > 1$?

Write the odd perfect number $m=p^k a^2$ as a product of primes $$m = p^k {p_1}^{2a_1} \cdots {p_v}^{2a_v}.$$ (Note that it is known that $v \geq 9$ by work of Nielsen.) Let $N(m)$ be the number of ...
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### Help with R. Ryan's "A simpler dense proof regarding the abundancy Index."

I'm reading Richard Ryan's article "A simpler dense proof regarding the abundancy index" and got stuck in his proof for Theorem 2. The Theorem is stated as follows: Suppose we have a ...
1 vote
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### On odd perfect numbers $p^k m^2$ with special prime $p$ satisfying $m^2 - p^k = 2^r t$ - Part II

(Preamble: We have asked this same question in MSE two weeks ago, without getting any answers. We have therefore cross-posted it to MO, hoping that it gets answered here.) The topic of odd perfect ...
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### Can perfect numbers be seen $p$-adically?

It is well known that all even perfect numbers are of the form $N=(2^{q}-1).2^{q-1}$ with $M_{q}:=2^{q}-1$ a Mersenne prime. As the very defining property of such a perfect number is to fulfill the ...
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### Is there a similar formula like Ramanunjan's Eisenstein series identity for $\sum_{k=1}^{n-1}k^2 \sigma(k)\sigma(n-k)$?

This question is related to the last question about van der Pol's identity for the sum of divisors. In Touchard (1953) it is mentioned that the sum of divisors $\sigma(n)$ satisfies the following ...
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### Frey's elliptic curve and perfect numbers?

Let $E_n:y^2=x(x-\sigma(n)/2)(x+\sigma(n)/n)$ be a Frey-elliptic curve, where $\sigma$ denotes the sum of divisors of the natural number $n$. If $n$ is a perfect number ($\sigma(n)=2n$) then the $j$-...
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### Perfect numbers, Galois groups and a polynomial

Let $f(n,t) = \sum_{k=0}^{r-1} d_k t^k$ where $D_n = \{d_0=1,d_1,\cdots,d_{r-1}\}$ are all divisors of $n$. For instance $$f(28,t) = 28 t^{5} + 14 t^{4} + 7 t^{3} + 4 t^{2} + 2 t + 1$$ For even ... 228 views

### Inductively computing Mersenne primes / perfect numbers?

For two sets $A,B$ set $A+B = \{a +b | a \in A,b \in B\}$. Let $(x_n)_{n \in \mathbb{N}}$ be independent variables. Let $\sigma(n)$ be the sum of divisors of $n$. Set $\hat{\phi}(1) = \{x_1\}$ and ... 341 views

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### Odd perfect numbers having as prime factors exclusively Mersenne primes and Fermat primes

I don't know if the following question is in the literature, please add a commment if it is in the literature. I add my thoughts and motivation below in last paragraph, it is discursive and ...
1 vote
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### Bounds for two arithmetic functions, when one assumes that $n$ are odd perfect numbers

For an integer $n>1$ in this post we denote the Dedekind psi function as $\psi(n)=n\prod_{\substack{p\mid n\\p\text{ prime}}}\left(1+\frac{1}{p}\right)$ and the product of distinct primes dividing ...
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### Around a characterization for even perfect numbers, similar than Euclides-Euler theorem, in terms of totatives

In this post we denote the sum of divisors function as $$\sigma(n)=\sum_{1\leq d\mid n}d,$$ then an even perfect number is a positive integer $n\equiv 0\text{ mod }2$ for which $\sigma(n)=2n.$ As ...
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### What work can be done to study the solutions of $\varphi\left(x^{\sigma(x)}\sigma(x)^x\right)=2^{x-1} x^{3x-1}\varphi(x)$?

For integers $n\geq 1$ I denote the Euler's totient function as $\varphi(n)$ and the divisor function $\sum_{1\leq d\mid n}d$ as $\sigma(n)$, that are two well-known mulitplicative functions. We ...
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### Attempt of exploit the equation $1/\operatorname{rad}(n)=1/2-2\varphi(n)/\sigma(n)$ in the context of even perfect numbers, and a related conjecture

It is well known that the problem concerning even perfect numbers is to prove or refute if there are infinitely many of them. Few weeks ago I wrote the following conjecture, where $\varphi(n)$ denotes ...
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### Perfect Runs of Consecutive Integers

A run of 2 or more consecutive integers is said to be perfect if the sum of its terms equals the sum of all the their proper divisors, adding common divisors as often as they occur. For example, 672, ...
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### A conjecture regarding odd perfect numbers

(Note: I asked this question in MSE this June 2018 but did not receive any responses there. I have therefore cross-posted it here, hoping that it gets answered.) Let $\sigma(z)$ denote the sum of ...
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### Perfect numbers and perfect powers

This was asked earlier at MSE. The observation that 28 = 27 + 1 shows that it is possible to have consecutive perfect numbers and perfect powers. However, this must be extremely rare. Is it ...
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### does infinite leinster groups indicates infinite perfect numbers?

The simplest definition i managed to find is: Leinster group is a finite group which her order equal the sum of its normal subgroups order, and as a perfect number is a positive integer that is equal ...
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### Conjecture on Odd Perfect Numbers

I'm a grad student in mathematics and I've been working with a very gifted high school student (likely the smartest high school student I've ever met) on problems he's brought up and some competition ...
(Note: This question is closely related to this other one in MSE.) Let $N = q^k n^2$ be an odd perfect number. From this paper in NNTDM, we have the equation $$i(q) := \frac{\sigma(n^2)}{q^k}=\frac{... 2 votes 1 answer 469 views ### If q^k n^2 is an odd perfect number with Euler prime q, is \sigma(q^k)/n + \sigma(n)/q^k bounded from above? STATEMENT OF THE PROBLEM If q^k n^2 is an odd perfect number with Euler prime q, is \sigma(q^k)/n + \sigma(n)/q^k bounded from above? MOTIVATION Let \sigma=\sigma_{1} denote the classical ... 0 votes 1 answer 225 views ### Can k be arbitrarily large in the following equations? (Note: This question was cross-posted from MSE per Dris's request.) Let N = q^k n^2 be an odd perfect number in Eulerian form. (That is, q is prime with q \equiv k \equiv 1 \pmod 4 and \gcd(q,... 6 votes 1 answer 1k views ### Kindda-Perfect number: Is there a sequence of numbers which are equal to the sum of its proper divisors excluding itself as well as 1? Perfect number is a positive integer that is equal to the sum of its proper divisors. The smallest perfect number is 6, which is the sum of 1, 2, and 3. Is there a sequence of numbers which are equal ... 1 vote 0 answers 130 views ### Is there an integer r \neq q (with r>1) such that N = q^k n^2 = \frac{r(r+1)}{2}\cdot{d} is an odd perfect number with d>1? Slowak showed in 1999 that every odd perfect number N = q^k n^2 can be written in the form$$N = \dfrac{{q^k}\sigma(q^k)}{2}\cdot{D}$$where D>1. From this result, it follows that every odd ... 0 votes 1 answer 147 views ### Proving k = 1 \implies q = 5, if q^k n^2 is an odd perfect number with Euler prime q Let N = q^k n^2 be an odd perfect number with Euler prime q. We want to show that the biconditional k = 1 \iff q = 5 holds. It suffices to prove one direction, as the implication q = 5 \... 3 votes 0 answers 165 views ### Looking for an appropriate reference(s) for two conjectures on odd perfect numbers (I apologize in advance if this question is unsuitable for MO. If so, please let me know and I will migrate it to MSE.) Let \sigma(M) be the sum of the divisors of the positive integer M. For ... 1 vote 0 answers 113 views ### If N = q^k n^2 is an odd perfect number, and n < q^{k+1}, does it follow that k > 1? Let N = q^k n^2 be an odd perfect number with Euler prime q. According to Dickson (as pointed out recently by Beasley), Descartes conjectured k=1 in a letter to Mersenne in 1638, with Frenicle'... 2 votes 0 answers 188 views ### On the conjectured nonexistence of even almost perfect numbers (other than powers of two) and odd perfect numbers [closed] (Note: This question has been cross-posted to MSE.) Let \sigma(a) = \sigma_{1}(a) be the sum of the divisors of the positive integer a. A number M is called almost perfect if \sigma(M) = 2M -... 2 votes 1 answer 236 views ### On attempting a proof for r > 1, if M = {2^r}{b^2} is an even almost perfect number which is not a power of two (Preamble: I first thought that this question might be more appropriate for MSE. However, I posted it here nonetheless in the hope that someone with that brilliant idea can help with answering my ... 1 vote 1 answer 313 views ### Is it possible to have an even superperfect number and an odd superperfect number whose product is an almost perfect number? A number n \in \mathbb{N} is said to be superperfect if$$\sigma(\sigma(n)) = 2n.$$A number m \in \mathbb{N} is said to be almost perfect if$$\sigma(m) = 2m - 1. Here is my question: Is ...
(Note: This question is an improved version of and has been cross-posted from this MSE post.) Let $\sigma(x)$ denote the sum of the divisors of $x$. If $\sigma(x) = 2x - 1$, then we call $x$ an ...