Questions tagged [divisor-sum]
For questions on the divisor sum function and its generalizations.
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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
We denote as $N$ an odd perfect number, and $d\mid N$ one of its divisors. We denote the Carmichael function as $\lambda(x)$, Wikipedia has the article Carmichael function dedicated to this number ...
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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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Reference request regarding odd 4-perfect numbers
I am reading a paper by Broughan and Zhou (2006) that is dealing with odd 4-perfect numbers. The title of the paper is "Odd multiperfect numbers of abundancy four."
In this paper they ...
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Largest possible prime factor for given $k$?
Let $k$ be a positive integer.
What is the largest possible prime factor of a squarefree positive integer $\ n\ $ with $\ \omega(n)=k\ $ (That is, it has exactly $\ k\ $ prime factors) satisfying the ...
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On the equation $\sigma\left(\square\right)=\text{prime}$: propositions that can be potentially interesting or reference request
For integers $A,B\geq 1$ we define the difference $\sigma(A)\sigma(B)-\sigma(AB)$, denoting it as $[A,B]$, where $\sigma(n)=\sum_{1\leq d\mid n}d$ denotes the sum of divisors function. It is possible ...
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How can we show this relationship between the recursive sum of divisors function and Figurate Number Polynomial on Primes?
Let us define the following recursive function involving the sum of divisors function $\sigma(n)$:
\begin{array}{ l }
r(n,1)=\sigma(n) \\
r(n,2)=\sum_{d|n}r(d,1) \\
r(n,3)=\sum_{d|n}r(d,2) \\
\...
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Are there infinite many squarefree numbers with $\ \varphi(n)\mid \sigma(n)\ $?
This question is inspired by this post.
I wonder whether the number of squarefree integers $\ n\ge 2\ $ with $\ \varphi(n)\mid \sigma(n)\ $ is still infinite. As in the link , $\ \varphi(n)\ $ is the ...
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Are there infinite many positive integers $\ n\ $ satisfying $\ \varphi(n)|\sigma(n)\ $?
Today I suddenly discovered that many positive integers $\ n\ $ satisfy $\ \varphi(n)\mid \sigma(n)\ $. This leads to the following question :
Are there infinitely many postive integers $\ n\ $ ...
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On odd perfect numbers and a GCD - Part VIII
(This question is an offshoot of earlier posts with a similar title and this recent preprint.)
Let $N = q^k n^2$ be an odd perfect number with special/Eulerian prime $q$ satisfying $q \equiv k \equiv ...
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Why does this identity with the product $\prod_{p\mid n}(p+k)$ and recursive sum of divisors function is true?
Let us define the following recursive function involving the sum of divisors function $\sigma(n)$:
\begin{array}{ l }
r(n,1)=\sigma(n) \\
r(n,2)=\sum_{d|n}r(d,1) \\
r(n,3)=\sum_{d|n}r(d,2) \\
\...
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How can we show this relationship between the sum of divisors function and the sum $p^{m}+2p^{m-1}+3p^{m-2}+\ldots+(m+1)$?
The sum of divisors function is commonly denoted by $\sigma(n)$. Now let us introduce a recursive definition of divisor functions:
$r_{n,1}=\sigma(n)$
$r_{n,2}=\sum_{d|n}r_{d,1}$
$r_{n,3}=\sum_{d|n}...
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If $s = 4m - 3$, then $\sigma(p^s) = (1 + p^{2m-1})(1 + p + \ldots + p^{2m-2})$. Is there a similar factorization for $\sigma(q^t)$ when $t=4n$?
Let $p,q$ be (odd) primes, and let $m,n,s,t$ be positive integers.
Denote the classical sum of divisors of the positive integer $x$ by $\sigma(x)=\sigma_1(x)$.
M. A. Nyblom showed that, if $s = 4m - 3$...
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Why is this alternating summation involving reciprocals of divisors always positive? A conjecture.
Conjecture. For all $n,m \in \Bbb{N}$,
$$
f(n, m) := \sum_{c\mid d\mid n\# \\ \gcd(c, 2m) = 1}\dfrac{(-1)^{\omega(d)}}{d}
$$
is greater than $0$.
Example verification code:
...
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Does $\sigma(m^2)/p^k$ divide $m^2 - p^k$, if $p^k m^2$ is an odd perfect number with special prime $p$?
The following query is an offshoot of this post 1 and this post 2.
Denote the classical sum of divisors of the positive integer $x$ by $\sigma(x)=\sigma_1(x)$.
The topic of odd perfect numbers likely ...
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What are the remaining cases to consider for this problem, specifically all the possible premises for $i(q)$?
Denote the classical sum of divisors of the positive integer $x$ by $\sigma(x)=\sigma_1(x)$. (Note that the divisor sum $\sigma$ is a multiplicative function.)
A number $P$ is said to be perfect if $\...
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Is this divisors related sequence asymptotically linear?
In order to study this Riemann hypothesis equivalent here, I have been looking at the first differences of this sequence:
$$a(n) = \sum _{m=1}^{n^2} \left(\sum _{k=1}^{n^2} [k|m][k\leq n][m\leq k \...
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Does there exist a nontrivial prime power $q^k$ such that $\sigma(n^2)/n = q^k$ for some $n$?
Let $\sigma(x)=\sigma_1(x)$ be the classical sum of divisors of the positive integer $x$.
My question in the present post is closely related to this one in MO:
QUESTION
Does there exist a nontrivial ...
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R.T.P. σ(p^n)=(p^(n+1)-1)/(p-1). (where σ denotes the divisor function) For a prime p. [closed]
I need to prove that for a prime p, σ(p^n) is (p^(n+1)-1)/(p-1).
E.g.
σ(3^3) is (3^(3+1)-1)/(3-1).
=3^(4)-1/2
=80/2
=40
Therefore σ(3^3)=40.
Let me know for any suggestions,proofs or references.
user1002440
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On odd perfect numbers and a GCD - Part VII
(Pardon me for being somewhat stubborn, but this question will be the last for this week. This post is an offshoot of this one.)
Let $N = q^k n^2$ be an odd perfect number be an odd perfect number ...
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Does $G \mid I$ and $I \mid H$ still hold if $\sigma(q^k)/2$ is not squarefree, where $q^k n^2$ is an odd perfect number with special prime $q$?
This question is an offshoot of this post #1 and this post #2.
Let $N = q^k n^2$ be an odd perfect number be an odd perfect number with special prime $q$ satisfying $q \equiv k \equiv 1 \pmod 4$ and $\...
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Please check my proof: $\gcd(\sigma(q^k),\sigma(n^2)) \mid \gcd(n^2,\sigma(n^2))$ holds if $q^k n^2$ is an odd perfect number with special prime $q$.
Let $q^k n^2$ be an odd perfect number with special prime $q$ satisfying $q \equiv k \equiv 1 \pmod 4$ and $\gcd(q,n)=1$.
It is known that
$$\gcd\bigg(\sigma(q^k),\sigma(n^2)\bigg)\cdot\gcd\bigg(n^2,\...
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Please check my proof: If $q^k n^2$ is an odd perfect number with special prime $q$, then $\sigma(q^k)/2 \mid n$ if and only if $n \mid \sigma(n^2)$.
Let $N = q^k n^2$ be an odd perfect number with special prime $q$ satisfying $q \equiv k \equiv 1 \pmod 4$ and $\gcd(q,n)=1$.
Denote the classical sum of divisors of the positive integer $x$ by $\...
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(Applied to OPN) Let $c$ and $d$ be integers, not both equal to zero. If $q$ and $r$ are integers such that $c = dq + r$, then $\gcd(c,d)=\gcd(d,r)$.
I was reading the following web resource, and found the following Lemma 8.1, which I think would be immensely useful for a computational task on odd perfect numbers (hereinafter abbreviated as OPN):
...
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If $km$ is a Descartes number with quasi-Euler prime $m$, must $\sqrt{k}$ be a squarefree palindrome?
Let $$\sigma(x) = \sum_{d \mid x}{d}$$ denote the sum of divisors of $x \in \mathbb{N}$, where $\mathbb{N}$ is the set of natural numbers or positive integers.
Recall that a Descartes number is an odd ...
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The map $m \to m\sigma(m)$ is not injective.
Let
$$\tau(n) = \sum_{d \mid n}{1}$$
be the divisor function,
$$\omega(n) = \sum_{p \mid n}{1}$$
be the prime divisor function,
$$\varphi(n) = \#\{1 \leqslant k \leqslant n : \gcd(k,n) = 1\}$$
be ...
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Does $\frac{\sigma(q^k)}{n} + \frac{\sigma(n)}{q^k} < 10$ hold in general for an odd perfect number $q^k n^2$ with special prime $q$?
This question is an offshoot of this MSE answer.
Let $\sigma(x)$ be the sum of the divisors of the positive integer $x$. (Denote the abundancy index of $x$ by $I(x)=\sigma(x)/x$.)
If $\sigma(M) = 2M$...
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sum of unbiased dice
Two unbiased dice are rolled and the numbers obtained are added. In this way, you have a better chance of getting numbers whose sum is:
a} 11
b} 10
c} 9
d} 8
e} 7
I did my sums and it is giving ...
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On the reciprocal of $I(n^2) - \frac{2(q - 1)}{q}$, if $q^k n^2$ is a(n) (odd) perfect number with special prime $q$
Denote the classical sum of divisors of the positive integer $x$ by $\sigma(x)=\sigma_1(x)$. Denote the abundancy index of $x$ by $I(x)=\sigma(x)/x$.
A number $N$ is said to be perfect if $\sigma(N)=...
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Does the inequality $I(n^2) \leq 2 - \frac{5}{3q}$ improve $I(q^k) + I(n^2) < 3$, 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 $q \equiv k \equiv 1 \pmod 4$ and $\gcd(q,n)=1$.
Denote the abundancy index of the positive integer $x$ by $I(x)=\sigma(x)/...
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Bounds for the abundancy index of divisors of odd perfect numbers in terms of the deficiency function - Part II
This post is an offshoot of this MSE question.
Motivation
Let $x, y$ and $z$ be positive integers.
Denote the sum of divisors of $x$ by $\sigma(x)$. Also, denote the deficiency of $y$ by $D(y)=2y-\...
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An equation about divisor function [closed]
We define the 2th divisor function as follows: $\sigma_2(n)=\sum_{d\mid n}d^2$?
Is there infinite positive integer $n$ such that $\sigma_2(n)=(n+3)^2$? If there is, how can we find the least $n$?
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If $\gcd(n^2, \sigma(n^2)) = q\sigma(n^2) - 2(q - 1) n^2$, does it follow that $q n^2$ is perfect?
Let $q$ be an (odd) prime, and let $\gcd(q,n)=1$.
Denote the classical sum of divisors of the positive integer $x$ by $\sigma(x)=\sigma_1(x)$.
A number $N$ is said to be perfect if $\sigma(N)=2N$.
...
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On the quantity $I(q^k) + I(n^2)$ considered as functions of $q$ and $k$, when $q^k n^2$ is an odd perfect number with special prime $q$ - Part II
Hereinafter, let $N = q^k n^2$ be an odd perfect number with special/Euler prime $q$ satisfying $q \equiv k \equiv 1 \pmod 4$ and $\gcd(q,n)=1$.
Denote the abundancy index of the positive integer $x$ ...
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On the conjectured inequality $q > k$, 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 with special/Eulerian prime $q$ satisfying $q \equiv k \equiv 1 \pmod 4$ and $\gcd(q,n)=1$.
Descartes, Frenicle, and subsequently Sorli conjectured that $k=1$ ...
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On odd perfect numbers and a GCD - Part VI
(Note: This post is closely related to this earlier MSE question.)
Let $N = q^k n^2$ be an odd perfect number with special/Eulerian prime $q$ satisfying $q \equiv k \equiv 1 \pmod 4$ and $\gcd(q,n)=1$....
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On odd perfect numbers and a GCD - Part V
(Note: This post is tangentially related to this earlier MSE question.)
Let $N = q^k n^2$ be an odd perfect number with special/Eulerian prime $q$ satisfying $q \equiv k \equiv 1 \pmod 4$ and $\gcd(q,...
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Additive divisor problems
There is plenty of literature on the binary additive divisor problem, that's evaluating
\[ \sum _{n\leq x}d(n)d(n+h)\]
for various $h$. Why is there nothing (or I don't find) on
\[ \sum _{n\leq x}d(n)...
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Help with "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 ...
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If $k=1$, then the index of an odd perfect number $p^k n^2$ (as defined by Chen and Chen (2014)) is not squarefree.
In what follows, we let $\sigma=\sigma_1$ denote the classical sum-of-divisors function. Denote the deficiency function of the positive integer $x$ by $D(x)=2(x)-\sigma(x)$, and the aliquot sum of $x$...
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Is it true that $l_1(q,n) \geq g(k)$, if $q^k n^2$ is an odd perfect number with special prime $q$?
(Note: This post is an offshoot of these earlier questions: (post 1) and (post 2).)
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(...
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On the inequality $I(q^k)+I(n^2) \leq \frac{3q^{2k} + 2q^k + 1}{q^k (q^k + 1)}$ where $q^k n^2$ is an odd perfect number
(Note: This post is an offshoot of this earlier MSE question.)
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 ...
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Is it possible to improve on these bounds for $\frac{\varphi(n)}{n}$, 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/Eulerian prime $q$ satisfying $q \equiv k \equiv 1 \pmod 4$ and $\gcd(q,n)=1$. Denote the Euler-totient function of the positive integer $x$ by ...
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If $q^k n^2$ is an odd perfect number with special prime $q$, is $\gcd(\sigma(q^k),\sigma(n^2))=1$ equivalent to $k=1$?
Let $q^k n^2$ be an odd perfect number with special prime $q$ satisfying $q \equiv k \equiv 1 \pmod 4$ and $\gcd(q,n)=1$.
Here is my:
QUESTION
If $q^k n^2$ is an odd perfect number with special prime ...
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Is $\limsup_n \frac{\sigma(n)}{n \log p(n)} <\infty$, where $p(n)$ is the greatest prime factor of $n$ and $\sigma(n)=\sum_{d | n} d$?
Let $\sigma(n)=\sum_{d | n} d$ and $p(n)$ be the greatest prime factor of $n$.
Can we prove that
$$\limsup_n \frac{\sigma(n)}{n \log p(n)} <\infty ?$$
I know that
$$\limsup_n \frac{\sigma(n)}{n \...
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A proposed unconditional factor-chain proof for the inequality $I(n) > \frac{3}{2}$, where $q^k n^2$ is an odd perfect number with special prime $q$
Let $N = q^k n^2$ be a hypothetical odd perfect number given in Eulerian form. (That is, $q$ is the special/Euler prime satisfying $q \equiv k \equiv 1 \pmod 4$ and $\gcd(q,n)=1$.)
Denote the ...
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Correctness of $\sigma_s(n) (1-\frac{1}{n^s}) = 0 \pmod{k}$ (divisor sum congruence)
I've played around with some identities and came up with:
$\sigma_s(n)(1-\frac{1}{n^s}) = 0 \pmod{k}$
(for $n$ and $k$ positive integers, and $s$ an integer)
With the conditions that 1) $n$ is ...
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On odd perfect numbers $q^k n^2$ and the deficient-perfect divisor $q^{\frac{k-1}{2}} n^2$ - Part III
(Preamble: This question is an offshoot of this answer by mathlove to an earlier post.)
Let $\sigma(x)=\sigma_1(x)$ denote the classical sum of the divisors of the positive integer $x$.
If $\sigma(m)=...
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On improving the upper bound $I(m^2) \leq \frac{2p}{p+1}$, if $p^k m^2$ is an odd perfect number with special prime $p$
(Preamble: This question is an offshoot of this earlier MSE post.)
Let $p^k m^2$ be an odd perfect number (OPN) with special prime $p$ satisfying $p \equiv k \equiv 1 \pmod 4$ and $\gcd(p,m)=1$.
...
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