Questions tagged [divisor-sum]
For questions on the divisor sum function and its generalizations.
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Is this proof for the divisibility constraint $\sigma(q^k)/2 \mid n$ correct, where $q^k n^2$ is an odd perfect number with special prime $q$?
Let $N = q^k n^2$ be an odd perfect number given in Eulerian form, i.e. $q$ is the special prime satisfying $q \equiv k \equiv 1 \pmod 4$ and $\gcd(q,n)=1$.
Define the GCDs
$$G = \gcd(\sigma(q^k),\...
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On the equations $D(q^k)D(n^2)=2s(q^k)s(n^2)$ and $\sigma(q^k)\sigma(n^2)=2 q^k n^2$, where $q^k n^2$ is an odd perfect number with special prime $q$
MOTIVATION
The topic of odd perfect numbers likely needs no introduction.
In what follows, we denote the classical sum of divisors of the positive integer $x$ by $\sigma(x)=\sigma_1(x)$, the ...
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Solution of $\sigma(\sigma(m)+m)=2\sigma(m)$ with $\omega(m)>8$?
This question is related to this one.
$\sigma(n)$ is the divisor-sum function (the sum of the positive divisors of $n$) and $\omega(n)$ is the number of distinct prime factors of $n$.
The object is ...
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Lower bound for divisor counting function
Let,
$$\tau(n)=\sum_{d|n}1$$
Be the divisors counting function.
Then is it true that,
There exists infinitely many $n$ satisfying,
$$\tau(n)>\left(\ln(n)\right)^{a}$$
Where $a\in[1,\infty)$?
My ...
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Is $n=2$ the only even solution of $\sigma(\sigma(n)+n)=2\sigma(n)$?
Inspired by this
question.
For positive integer $m$ , let $\sigma(m)$ be the divisor-sum function.
Let $S$ be the set of positive integers $n$ satisfying $$\sigma(\sigma(n)+n)=2\sigma(n)$$ In the ...
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Is there an infinite composite numbers $n$ such that $\sigma(\sigma(n)+n) = 2 \cdot \sigma(n)$
For a positive integer $n$, let $\sigma(n)$ denote the sum of the positive divisors of $n$.
Now, suppose that $n$ is a composite number. Is there an infinitely many $n$ composite such that $\sigma(\...
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Revisiting MSE question 4386812 - Determining whether $\sigma(q^k)/2$ is squarefree, where $q^k n^2$ is an odd perfect number with special prime $q$
Preamble: The present inquiry is an offshoot of this earlier MSE question.
MOTIVATION
Denote the classical sum of divisors of the positive integer $x$ by $\sigma(x)=\sigma_1(x)$. (Note that the ...
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Is this disproof for the Descartes-Frenicle-Sorli Conjecture that $k=1$, if $p^k m^2$ is an odd perfect number, valid?
Let $N = p^k m^2$ be an odd perfect number with special prime $p$ satisfying $p \equiv k \equiv 1 \pmod 4$ and $\gcd(p,m)=1$.
It is known that
$$D(p^k)D(m^2)=2s(p^k)s(m^2) \tag{0}$$
where $D(x)=2x-\...
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Divisors count of $3^n\pm1$
During the investigation of $3^n\pm1$ visually saw that the divisors count of it are sums of $2^m$.
Especially if we take $n=p_1p_2$ ($p_i$ is prime) then at least for $n=p_1p_2<130$ it is either ...
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If $p^k m^2$ is an odd perfect number, then $D(p^k)/s(p^k)$ is in lowest terms. Does this contradict $D(p^k)D(m^2)=2s(p^k)s(m^2)$?
In what follows, denote the classical sum of divisors of the positive integer $x$ by $\sigma(x)=\sigma_1(x)$.
If $n$ is odd and $\sigma(n)=2n$, then $n$ is called an odd perfect number.
Euler showed ...
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Is there an analytical solution to the inequality $\frac{1 + k({p}^{(k+1)/2)})}{p^k} < \frac{p}{p - 1}$ if one were to bound $k$ in terms of $p$?
My question is as is in the title:
Is there an analytical solution to the inequality
$$\frac{1 + k({p}^{(k+1)/2)})}{p^k} < \frac{p}{p - 1}$$ if one were to bound $k$ in terms of $p$?
Here, $p \...
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Is there an example for every given exponent?
Inspired by this question , where it is asked for positive integers with the property $$2n-\sigma(n)\mid \sigma(n)-n$$ which is equivalent to $$2n-\sigma(n)\mid n$$ The author also demands $2n-\sigma(...
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Why is it that, if there are no odd perfect numbers, then there are no other $3$-perfect numbers, apart from the six known, as of the year $1643$?
Let $\sigma(x)=\sigma_1(x)$ denote the classical sum of divisors of the positive integer $x$.
A number $N$ is said to be $k$-perfect if $\sigma(N)=kN$ where $k$ is a positive integer.
The number $1$ ...
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Sum of Factors Question [duplicate]
Let $d_1,d_2,\ldots,d_k,$ be all the factors of a positive integer $n,$ including $1,$ and $n.$ Suppose $d_1+d_2+\ldots+d_k=72.$ Then, find the value of $\frac{1}{d_1}+\frac{1}{d_2}+\ldots+\frac{1}{...
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Euler product involving divisor function
Take $k,N\in \mathbb N$ and $s\in \mathbb C$ with real part $\sigma \in [1-\delta ,1]$ for some small fixed $\delta $. In its simplest form my question is how do I sum $$\sum _{l\geq 0}\frac {d_k(p^{...
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Define $\partial(n\mid x) = \partial(q_0 \cdots q_i \mid x) = \sum_{j=0}^i (-1)^i q_j (\frac{n}{q_j} \mid x)$. What does homology measure?
Let $R$ be a commutative ring with $1$ and let $M = \{ f : \Bbb{N} \to R \}$ be the $R$-module of arithmetic functions into $R$.
A basis for $M$ is $(d \mid \cdot) : d \in \Bbb{N}$ where $(d\mid n) = ...
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If $p$ is a prime number and $k$ is a positive integer, is it true that $\sigma_1(p^k) > 1 + k (\sqrt{p})^{1+k}$?
Denote the classical sum of divisors of the positive integer $x$ by $\sigma(x)=\sigma_1(x)$.
Here is my initial question:
If $p$ is a prime number and $k$ is a positive integer, is it true that
$$\...
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If $p^k m^2$ is an odd perfect number with special prime $p$ and $p = k$, then $\sigma(p^k)/2$ is not squarefree.
While researching the topic of odd perfect numbers, we came across the following implication, which we currently do not know how to prove:
CONJECTURE: If $p^k m^2$ is an odd perfect number with ...
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The mean square of $d_k(n)$
Let $d_2(n)=d(n)$ be the divisor function, and let $$d_k(n)=\sum_{d_1\cdots d_k= n}1=\sum_{m\cdot l= n}d_{k-1}(m).$$ Can anyone point me to a reference to the size of the error term when approximating
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Is this logical deduction regarding some modular restrictions on odd perfect numbers valid? - Part II
Let $p^k m^2$ be an odd perfect number with special prime $p$ satisfying $p \equiv k \equiv 1 \pmod 4$ and $\gcd(p,m)=1$. Denote the classical sum of divisors of the positive integer $x$ by $\sigma(x)...
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Is this logical deduction regarding some modular restrictions on odd perfect numbers valid?
Let $p^k m^2$ be an odd perfect number with special prime $p$ satisfying $p \equiv k \equiv 1 \pmod 4$ and $\gcd(p,m)=1$. Denote the classical sum of divisors of the positive integer $x$ by $\sigma(x)...
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On the prime factorization of $n$ and the quantity $J = \frac{n}{\gcd(n,\sigma(q^k)/2)}$, where $q^k n^2$ is an odd perfect number
Let $N = q^k n^2$ be an odd perfect number with special prime $q$ satisfying $q \equiv k \equiv 1 \pmod 4$ and $\gcd(q,n)=1$. Denote the classical sum of divisors of the positive integer $x$ by $\...
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On a consequence of $G \mid I \iff \gcd(G, I) = G$ (Re: Odd Perfect Numbers and GCDs)
Let $N = q^k n^2$ be an odd perfect number given in the so-called Eulerian form, where $q$ is the special prime satisfying $q \equiv k \equiv 1 \pmod 4$ and $\gcd(q,n)=1$. Denote the classical sum of ...
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Find all positive integers $n$ such that $\sigma(n) = n + 1 + \sigma(n+1)$
Find all positive integers $𝑛$ such that
$\sigma\left(n\right) = n + 1 + \sigma\left(n + 1\right)$, where $\sigma\,()$ is the divisor function.
I found $n = 18,\ 3200$.
For $n \leq 10^8$, only the ...
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Can this inequality involving the deficiency and sum of aliquot divisors be improved? - Part II
This MSE question (from April 2020) asked whether the inequality
$$\frac{D(n^2)}{s(n^2)} < \frac{D(n)}{s(n)}$$
could be improved, where $D(x)=2x-\sigma(x)$ is the deficiency of the positive integer ...
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Is $650$ the only solution not fitting in the family?
Inspired by this question
The linked question conjectures that $\frac{\sigma(n)}{n+1}$ (where $\sigma(n)$ denotes the divisor-sum function) is not an integer for any squarefree composite number.
If we ...
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On the equation $s(n^2) = \left(\frac{q-1}{2}\right)\cdot{D(n^2)}$, if $q^k n^2$ is an odd perfect number with special prime $q$
Let $N$ be an odd perfect number given in the so-called Eulerian form
$$N = q^k n^2$$
where $q$ is the special prime satisfying $q \equiv k \equiv 1 \pmod 4$ and $\gcd(q,n)=1$.
In what follows, let ...
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On Tony Kuria Kimani's recent preprint in ResearchGate
(Preamble: The method presented here to compute the GCD $g$ is patterned after the method used to compute a similar GCD in this answer to a closely related MSE question.)
Let $\sigma(x)=\sigma_1(x)$ ...
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Proof-verification request: On the equation $\gcd(n^2,\sigma(n^2))=D(n^2)/s(q^k)$ - Part II
(Preamble: This inquiry is an offshoot of this answer to a closely related question.)
In what follows, denote the classical sum of divisors of the positive integer $x$ by $\sigma(x)=\sigma_1(x)$, the ...
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If $2^r b^2$ is an even almost perfect number that is NOT a power of two, does it follow that $r=1$?
(The following are taken from this preprint by Antalan and Dris.)
Antalan and Tagle showed that an even almost perfect number $n \neq 2^t$ must necessarily have the form $2^r b^2$ where $r \geq 1$, $\...
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Does $\sigma(n^2)/q \mid q^k n^2$ imply $\sigma(n^2)/q \mid n^2$, if $q^k n^2$ is an odd perfect number with special prime $q$? - Part II
(Preamble: This inquiry is an offshoot of this MSE question.)
Denote the classical sum of divisors of the positive integer $x$ by $\sigma(x)=\sigma_1(x)$. Denote the abundancy index of $x$ as $I(x)=\...
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prime factorizations/ sum of squares of divisors.
Find all positive integers $n$ such that the sum of the squares of the divisors of $n$ is equal to $n^2+2n+37$, and in which $n$ is not of the form $p(p+6)$ where p and p+6 are prime numbers.
I tried ...
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Are these valid proofs for the equation $\gcd(n,\sigma(n^2))=\gcd(n^2,\sigma(n^2))$, if $q^k n^2$ is an odd perfect number with special prime $q$?
Let $N = q^k n^2$ be an odd perfect number with special prime $q$ satisfying $q \equiv k \equiv 1 \pmod 4$.
It is known that
$$i(q)=\gcd(n^2,\sigma(n^2))=\frac{\sigma(n^2)}{q^k}=\frac{n^2}{\sigma(q^k)/...
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Find all natural numbers $n$ such that the sum of the squares of the positive divisors of $n$ is equal to $n^2+2n+37$.
If $f(n)$ is the sum of the squares of the positive divisors of n,then find all natural numbers $n$ such that $f(n)=n^2+2n+37$. I tried it first by substituting few values of $n$ but that doesn't work ...
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On a curious PDE involving $I(q^k)+\frac{2}{I(q^k)}$
Let $I(x)=\sigma(x)/x$ be the abundancy index of the positive integer $x$, where $\sigma(x)=\sigma_1(x)$ is the classical sum of divisors of $x$.
Consider
$$f(q,k)=I(q^k)+\frac{2}{I(q^k)}$$
where $5 \...
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How did Ramanujan came up with this?
The following is a picture of equation from Ramanujan's lost notebook. In this page, Ramanujan gives a closed form for,
$$\sum_{n\geq 1}\sigma_{s}(n)x^{n}$$
In an attempt initially it's claimed that,
...
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Are the (Bezout) coefficients for $\gcd(n^2, \sigma(n^2)) = q\sigma(n^2) - 2(q - 1)n^2$ (where $q^k n^2$ is an odd perfect number) unique?
In what follows, denote the classical sum of divisors of the positive integer $x$ by $\sigma(x)=\sigma_1(x)$, the deficiency of $x$ by $D(x)=2x-\sigma(x)$, and the aliquot sum of $x$ by $s(x)=\sigma(x)...
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Is there any other known relationship between even perfect numbers and odd perfect numbers, apart from their multiplicative forms?
(Note: This was cross-posted from MO, because it was not well-received there. Will delete the MO post in a few.)
Observe that an even perfect number $M = (2^p - 1)\cdot{2^{p - 1}}$ and an odd perfect ...
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Particular values for the sum of divisors function from billiards
In this post we consider as reference the article Arithmetic billiards from Wikipedia. We consider the arithmetic billiard that is explained in the article.
I've wondered if we can compute some simple ...
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Finding a formula for $\sum_{d|n} \tau(d)$, where $\tau(d)$ is the number of divisors of $d$.
I am currently in the middle of the following exercise:
Exercise. Compute $$ \sum_{d|n} \left(\sigma(d)\mu\left(\frac{n}{d}\right)+\tau(d)\right),$$
where $\sigma$ is the function that corresponds to ...
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What are the most efficient ways to calculate the sum of positive divisors function, σ, and aliquot sum, s?
Given a positive integer $n$, what are the most efficient algorithms for calculating the sum of positive divisors function $\sigma_{1}(n)$ and the aliquot sum $s(n)$ ?
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Hand calculation of divisor summatory function
Maybe this question is stupid, but there was a problem in a math competition (not even in the highest stage) in my county which asked to
Find $ \sum_{n\leq390} d(n)$, where $d(n)$ is the number of ...
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If $q^k n^2$ is an odd perfect number with special prime $q$, is it possible to express $\sigma(n^2)/q^k$ as a function of only $q$ and $k$?
Let $q^k n^2$ be an odd perfect number with special prime $q$ satisfying $q \equiv k \equiv 1 \pmod 4$ and $\gcd(q,n)=1$. Denote the classical sum of divisors of the positive integer $x$ by $\sigma(x)...
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Does the following lower bound improve on $I(q^k)+I(n^2) > \frac{57}{20}$, where $q^k n^2$ is an odd perfect number?
Preamble: This question is an offshoot of this earlier post. (This inquiry has likewise been cross-posted to MO last June $10, 2022$.)
Let $N = q^k n^2$ be an odd perfect number with special prime $q$...
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Finding the Ceiling function of $(\sqrt{3}+ 1)^{2n}$
Ceiling function of $(\sqrt{3}+ 1)^{2n}$ is $(\sqrt{3} + 1)^{2n} + (\sqrt{3} - 1)^{2n}$.
While solving a problem that states $2^{n+1}$ divides ceiling function of $(\sqrt{3}+ 1)^{2n}$. I went through ...
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Follow-up to MSE question 3738458
This is a follow-up inquiry to this MSE question.
Let $\sigma(x)$ denote the sum of divisors of the positive integer $x$.
A number $M$ is said to be perfect if $\sigma(M)=2M$. For example, $6$ and $...
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What conditions on $X$ will guarantee that $\gcd(\text{square part of } X,\text{squarefree part of } X)=1$, if $X$ is neither a square nor squarefree?
The following query is an offshoot of this answer to a closely related post.
Denote the classical sum of divisors of the positive integer $x$ by $\sigma(x)=\sigma_1(x)$.
The topic of odd perfect ...
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Exponential sums with the divisor function
Can anyone explain the first $\ll $ on line 8 on page 188 of "Jutila: On exponential sums involving the divisor function" for me? (https://eudml.org/doc/152693)
Specifically, I think he is ...
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On Carmichael function and aliquot parts of odd perfect numbers
This post is cross-posted on MathOverflow with identifier 439563 and same title.
We denote as $N$ an odd perfect number, and $d\mid N$ one of its divisors. We denote the Carmichael function as $\...
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Does $I = \gcd(n,\sigma(n^2)) = (\frac{n}{\sigma(q^k)/2})\cdot\gcd(\sigma(q^k)/2,n)$ imply that $\sigma(q^k)/2 \mid n$ holds?
Let $N = q^k n^2$ be an odd perfect number with special prime $q$ satisfying $q \equiv k \equiv 1 \pmod 4$ and $\gcd(q,n)=1$.
Define the GCDs:
$$G = \gcd\bigg(\sigma(q^k),\sigma(n^2)\bigg)$$
$$H = \...
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