20 votes
Accepted

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?

Such an abundant number with abundance 1 is called a quasiperfect number (which is a more professional way to say "kindda-perfect"). None have been found, according to Wikipedia. This 1982 article ...
Carlo Beenakker's user avatar
19 votes

Perfect numbers and perfect powers

Using Joe Silvermann’s notations, and under the assumption that any perfect number is even, we are trying to find the integer solutions of $$2^{p-1}(2^p-1)=x^m+1,$$ where $x \geq 2, m \geq 2, p\geq 2$ ...
Mindlack's user avatar
  • 191
11 votes

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

Curiously enough, I asked myself the same question several days ago... I couldn't settle it; yet, resorting to Jacques Touchard's theorem on the form of odd perfect numbers (cf. J. A. Holdener, "...
José Hdz. Stgo.'s user avatar
9 votes

Is there a similar formula like Ramanunjan's Eisenstein series identity for $\sum_{k=1}^{n-1}k^2 \sigma(k)\sigma(n-k)$?

All these identities can indeed be proved essentially trivially using modular forms and quasi-modular forms (those involving $E_2$), and the fact that the dimension of such spaces is $1$ for weight 4,...
Henri Cohen's user avatar
  • 11.5k
9 votes
Accepted

On a GCD approach to odd perfect numbers

As I mentioned in another answer, using GCDs and fractions is usually a bad idea. Here is how I would interpret all of this material. Let $N$ be an odd perfect number. Write its prime factorization ...
Pace Nielsen's user avatar
8 votes

Perfect numbers, Galois groups and a polynomial

The second factor is $P(2t)$ where $P=X^{p-1}+\cdots+X+1$, the $p$-th cyclotomic polynomial. Hence the Galois group of $P(2t)$ is the same as the Galois group of $P(t)$, which is simply $(\mathbb{Z}/p\...
GreginGre's user avatar
  • 1,661
8 votes
Accepted

Is there a similar formula like Ramanunjan's Eisenstein series identity for $\sum_{k=1}^{n-1}k^2 \sigma(k)\sigma(n-k)$?

Numerical experiments suggest that $$A_2(n) := \sum_{k=1}^{n-1} k^2\sigma(k)\sigma(n-k) = \frac{n^2}{8}\sigma_3(n) - \frac{4n^3-n^2}{24}\sigma(n).$$ PS. In fact, it directly follows from the quoted ...
Max Alekseyev's user avatar
8 votes
Accepted

Can perfect numbers be seen $p$-adically?

(Not a complete answer but a bit too long for a comment.) There's a fundamental difficulty here in proving the sort of result you envision. If there were some prime $p$ which had to divide every odd ...
JoshuaZ's user avatar
  • 6,090
7 votes
Accepted

A possible axiomatic characterization of the set of divisors of a perfect number

Try this. $$ S = \{1,2,3,12,18,36\} $$ then $$ \sum_{k \in S}\frac{1}{k} = 2 \in S $$ and $$ \frac{1}{6}\sum_{k \in S} k \pi(k) = 36 \in S $$ when $\pi$ is the permutation of $S$ that reverses the ...
Gerald Edgar's user avatar
  • 40.2k
7 votes

Perfect numbers and perfect powers

If you accept the conjecture that there are no odd perfect numbers, then the finiteness follows quite easily from the $ABC$ conjecture. Thus an example has the form $$ 2^{p-1}(2^p-1) = x^m + 1. $$ ...
Joe Silverman's user avatar
6 votes

Generalized quasi-perfect numbers

See the abstract of my PhD thesis, "Generalised quasiperfect numbers", Bulletin Australian Math. Soc., 27 (1983), 153-156, where I consider numbers $n$ with $\sigma(n) = 2n + k^2$, $k$ odd, $(n,k)=1$. ...
Graeme Cohen's user avatar
6 votes

Perfect Runs of Consecutive Integers

523776, 523777 is another example of a 2-run. Again, 523776 is a triperfect number. An example of a 3-run is 5829840, 5829841, 5829842. No apparent pattern in these three numbers: ...
Freddy Barrera's user avatar
6 votes

A conjecture regarding odd perfect numbers

It is rare that complicating an expression leads to an insight. There are, of course, important counter-examples to this principle. But generally speaking, one seeks to simplify an expression, ...
Pace Nielsen's user avatar
6 votes
Accepted

Can an even perfect number be a sum of two cubes?

Here is a proof that 28 is the only even perfect number that is the sum of two positive cubes. The proof in Gallardo's article must be adapted in the case $x,a$ are even. Write $N=2^{p-1}(2^p-1) = x^3+...
François Brunault's user avatar
5 votes
Accepted

The action of the unitary divisors group on the set of divisors and odd perfect numbers

Here are some general comments: You don't need to bring these actions of abelian groups on various sets of divisors. The identity $$\sigma(n)=\sum_{d^2|n}d\sigma^{*}(\frac{n}{d^2})$$ is easy to check ...
Gjergji Zaimi's user avatar
4 votes
Accepted

Bounds for the number of prime numbers less than the Euler's factor, the radical and the greatest prime factor, respectively, of an odd perfect number

Since we have good asymptotics for $\pi(n)$ by the prime number theorem (and can get good explicit bounds on that from Rosser and Schoenfeld's work as well as later work such as that by Dusart) this ...
JoshuaZ's user avatar
  • 6,090
4 votes
Accepted

What work can be done to study the solutions of $\varphi\left(x^{\sigma(x)}\sigma(x)^x\right)=2^{x-1} x^{3x-1}\varphi(x)$?

Here is a proof that if an odd integer $x>1$ satisfies (1), then $x$ is a perfect number. First, by using the property that $\varphi(nm)=n\varphi(m)$ whenever $\mathrm{rad}(n)\mid\mathrm{rad}(m)$, ...
Max Alekseyev's user avatar
4 votes
Accepted

Steuerwald's theorem

Here is a proof of this fact. We start with a standard Lemma 1. Any prime divisor $q$ of $1+x+x^2$ for an integer $x$ is either equal to 3 or is congruent to 1 modulo $3$. Proof. If, on the contrary, ...
Fedor Petrov's user avatar
4 votes
Accepted

If $p^k m^2$ is an odd perfect number with special prime $p$, then must $m^2 - p^k = s^2 - t^2$ hold for some $s$ and $t$?

The answer is yes. Since $p\equiv 1\pmod{4}$, and $m$ is odd, we have $m^2-p^k\equiv 0\pmod{4}$. The only integers that are not differences of squares are exactly those that are $2\pmod{4}$.
Pace Nielsen's user avatar
3 votes

Conjecture on odd perfect numbers

The earliest result appears to be $$\frac{1}{2}<\sum_{p\mid N}\frac{1}{p}<2 \log \left (\frac{\pi}{2} \right ),$$ as proved by Perisastri. Other authors have sharpened this result. For example, ...
user118621's user avatar
3 votes

Generalized quasi-perfect numbers

The related equation $$\sigma(n) = An + B(n)$$ where $B(n)$ "is a function that may depend on properties of $n$" is considered in the paper Variations on Euclid’s Formula for Perfect Numbers by ...
Jose Arnaldo Bebita Dris's user avatar
3 votes
Accepted

Proving $k = 1 \implies q = 5$, if $q^k n^2$ is an odd perfect number with Euler prime $q$

The equivalence : $(k=1 \lor q> 5) \iff (q=5 \implies k=1)$ is not correct on its own, except if you have some other information (I don't know the problem, I'm simply evaluating the "logical" ...
Maxtimax's user avatar
  • 180
3 votes

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

Not an answer, but I just want to point out some thoughts that recently occurred to me, which are related to this problem. By this answer, we know that every odd perfect number $N = q^k n^2$ can be ...
Jose Arnaldo Bebita Dris's user avatar
3 votes
Accepted

Inductively computing Mersenne primes / perfect numbers?

The conjecture fails for $n=8128$, which can be verified in matter of seconds as explained below. I used PARI/GP for my verification. First, since the conjecture concerns only values of at $x$'s being ...
Max Alekseyev's user avatar
3 votes

Bounds for two arithmetic functions, when one assumes that $n$ are odd perfect numbers

As far as I'm aware, we don't have any substantially non-trivial bounds on the behavior of $\psi(n)$ when $n$ is an odd perfect number. We can at least prove the following but none of these are ...
JoshuaZ's user avatar
  • 6,090
3 votes
Accepted

On the nearest-square function and the quantity $m^2 - p^k$ where $p^k m^2$ is an odd perfect number

Middle of page 6 of https://arxiv.org/pdf/1312.6001v10.pdf " we always have $0 < n−\lceil\sqrt{n^2−q^k}\rceil$ " No, this requires that $q^k\ge 2n-1$, an helpful assumption when the goal ...
Pascal Ochem's user avatar
3 votes
Accepted

Number of distinct near-squares primes dividing an odd perfect number

In general, very few prime factors in an odd perfect number can be of the form $n^2+1$. In particular, if N is an odd perfect number then $\frac{\sigma(N)}{N}=2$, and for any $m$ (perfect or not), $\...
JoshuaZ's user avatar
  • 6,090
3 votes
Accepted

Divisibility relation with a specific sum of divisors

From the paper of Touchard that is linked in the question on we get the relation $$3nS_0(n)-\frac{n(n-1)\sigma(n)}{6}=\frac{10}{n}S_2(n) ....(1)$$ here $S_i(n)=\sum_{k=1}^{n-1}k^i\sigma(k)\sigma(n-k)$....
Alapan Das's user avatar
  • 1,755
3 votes
Accepted

On the largest prime factor and the largest component of an odd perfect number

Not a complete or as organized as an answer should be, too long for a comment. (Also I suspect that most of what I write here is going to be stuff you already know.) One naïve approach here is to try ...
JoshuaZ's user avatar
  • 6,090
2 votes
Accepted

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

This should be provable by standard although laborious methods. What follows is a proof sketch (I have not checked all the computational details but this method should work). We recall a few basic ...
JoshuaZ's user avatar
  • 6,090

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