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.

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Has it been proved that odd perfect numbers cannot be triangular?

(Note: This question has been cross-posted to MO.) Euclid and Euler proved that every even perfect number is of the form $m = \frac{{M_p}\left(M_p + 1\right)}{2}$ where $M_p = 2^p - 1$ is a prime ...
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Is there a notion “above” that of perfect numbers?

When trying to understand a notion, it often gives great insight to see it as the "shadow" of something bigger, carrying more information. The notion of categorification relies on this idea. A basic ...
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If we suppose, that a perfect odd number exists, can we determine whether infinite many exist?

It is currently not known whether odd perfect numbers (numbers with the property $\sigma(n)=2n$, where $\sigma(n)$ is the sum of divisors of $n$, including $1$ and $n$) exist. But suppose, a perfect ...
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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 ...
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If $L > 1$ is an odd almost perfect number with $\omega(L)=6$, then $L$ must be divisible by $3$.

Edited July 15 2016 Let $\mathbb{N}$ denote the set of positive integers. Let $\sigma = \sigma_{1}$ denote the (classical) sum-of-divisors function. Let $I(x) = \dfrac{\sigma(x)}{x}$ denote the ...
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On some inequalities relating the special/Euler prime and non-Euler part of odd perfect numbers

Let $N$ be an odd (positive) integer. If $\sigma(N)=2N$ where $\sigma(N)$ is the sum of the divisors of $N$, then $N$ is called an odd perfect number. Let $I(N)=\sigma(N)/N$ denote the abundancy ...
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On Basak's “Bounds On Factors Of Odd Perfect Numbers”

Let $N = q^k n^2$ be an odd perfect number given in Eulerian form, i.e. $q$ is the special/Euler prime satisfying $q \equiv k \equiv 1 \pmod 4$ and $\gcd(q,n)=1$. In what follows, we denote the ...
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Is there an anomaly in the distribution of Mersenne primes?

In a recent press release off the Great Internet Mersenne Prime Search distributed computing project page, it is announced that $$2^{82589933} - 1$$ is the largest known (Mersenne) prime, ...
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Can an odd perfect number be divisible by $119$?

We know that an odd perfect number cannot be divisible by $105$ (see here), by $825$ (see here), and by $5313$ (see here). I wonder if that's also the case for $119$.
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On the indivisibility of odd perfect numbers by small numbers

A good day to everyone! This question is an offshoot of the following MSE posts: Odd perfect number divisors Can an odd perfect number be divisible by $101$? My question is as follows: Is ...
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What do we know about Sorli's Conjecture and Odd Perfect Numbers?

I have seen that Jose Arnaldo Dris has recently published a great deal of literature concerning odd perfect numbers and Sorli's Conjecture. He claims that the truth of Sorli's conjecture implies that ...
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Combinatorial interpretation of Euclid's form for even perfect numbers

Euclid showed that if $p$ is a prime such that $2^{p}-1$ is also a prime, then the number $n=2^{p-1}.(2^{p}-1)$ is perfect. Much later, Euler proved that every even perfect number is of this form. ...
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Did Descartes and Frénicle consider deficient-perfect numbers?

Holdener and Rachfal show that, if $N = q^k n^2$ is an odd perfect number with special prime $q$ (satisfying $q \equiv k \equiv 1 \pmod 4$ and $\gcd(q,n)=1$), then $q^{(k-1)/2} n^2$ is deficient-...
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Does this iterated sequence always end in a finite number of steps to a number which is divisible by a perfect number?

I posted this question at MathOverflow, but then I realized that maybe it is more appropriate to ask it here: Let $f$ be a multiplicative arithmetic function which maps $\mathbb{N}$ to itself, such ...
Let $N=q^k n^2$ be an odd perfect number with special prime $q$. The index $i(q)$ of $N$ at the prime $q$ is then equal to $$i(q):=\frac{\sigma(n^2)}{q^k}=\frac{n^2}{\sigma(q^k)/2}=\frac{D(n^2)}{s(q^... 0answers 26 views A consequence of assuming the Descartes-Frenicle-Sorli Conjecture on odd perfect numbers Let x be a positive integer. Denote the sum of divisors of x by$$\sigma(x) = \sum_{d \mid x}{d},$$and the deficiency of x by$$D(x) = 2x - \sigma(x).$$A number N is said to be perfect if ... 0answers 24 views Divisors of a  k -multiperfect number Let  n  be a  k -multiperfect number. Denote by  d_m  its  m  smallest divisor, and  n_{m}  the number of divisors of  n  divisible by  d_m . Is there for all  2\leq m\leq\tau(n) ... 0answers 20 views Has it been conjectured that all k-multiperfect numbers are multiples of k? A quick glance at the list of the first  k  -multiperfect numbers for small  k  makes me think that all  k  -multiperfect numbers are multiples of  k  , which is a generalization of the ... 0answers 83 views Does mathlove's answer imply that D(n^2) \mid n^2? In what follows, set \sigma(x) to be the sum of divisors of x \in \mathbb{N}, and let$$D(x) = 2x - \sigma(x)$$be the deficiency of x, and let$$s(x) = \sigma(x) - x$$be the sum of the aliquot ... 0answers 50 views Specific evidence for or against the Descartes-Frenicle-Sorli conjecture on odd perfect numbers This answer to an earlier (and related) MSE question summarizes what appears to be the first documented "evidence" against the Descartes-Frenicle-Sorli conjecture that k=1, if q^k n^2 is an odd ... 0answers 35 views If n = km is a Descartes number with quasi-Euler prime m, then m < k. Let$$\sigma(x) = \sum_{d \mid x}{d}$$denote the sum of divisors of x \in \mathbb{N}, where \mathbb{N} is the set of natural numbers or positive integers. Recall that a Descartes number is an ... 0answers 73 views Bounds for an expression involving the divisors of an odd perfect number Let$$\sigma(x) = \sum_{l \mid x}{l}.$$That is, let \sigma(x) denote the sum of the divisors of x \in \mathbb{N} (the set of natural numbers or positive integers). Set$$D(x) := 2x - \sigma(x)$$... 0answers 239 views What is the relation between the perfect squares and prime numbers? \hspace{10pt}I was playing with R and a list of the first 10,000 primes and suddenly I had the idea of looking how many perfect squares are between p_n and p_{n+1}. \hspace{10pt}I make a ... 0answers 174 views Conjecture: a^3 + b^3 + c^3 = p^3 \Rightarrow x^3 + y^3 + z^3 = \big(\frac{a + b + c}{2}\big)^3 Conjecture: Let p be an even perfect number, and a, b, c be positive natural numbers. There exists values for a, b, c to satisfy the following equation$$a^3 + b^3 + c^3 = p^3 for ...
So I have made a theorem: You are given two prime numbers $P_n$ and $P_{n + 1}$ such that $P_n$ denotes the $n^{th}$ prime number and $P_{n + 1}$ is the following prime number from $P_n$. Take ...