Questions tagged [divisor-sum]
For questions on the divisor sum function and its generalizations.
612
questions
3
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Prove that $ a_{1}^{3}+a_{2}^{3}+\cdots+a_{l}^{3}=\left(a_{1}+a_{2}+\cdots+a_{l}\right)^{2} $
Let $ d_{1}, d_{2}, \ldots, d_{l}$ be all positive divisors of a positive integer $n$. For
each $i=1,2, \ldots, l$ denote by $a_{i}$ the number of positive divisors of $d_{i}$. Then
$
a_{1}^{3}+a_{2}^{...
0
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0answers
34 views
Polynomial properties
I am interested in finding out the name of the following polynomials (if already named) and additionaly any of their properties:
$$p_N(x) = \sum_{d|N}{x^{d+\frac{N}{d}}}$$
It is clear they will only ...
5
votes
1answer
112 views
Do the numbers preceding primes have on an average fewer divisors than the numbers succeeding primes?
I wanted to see if the numbers preceding primes behaved differently in any way form the numbers succeeding primes so I calculated at the average number of divisors of number of the form $p-1$ and $p+1$...
3
votes
1answer
78 views
The summation $\sum_{n\geqslant1} \frac1n\sum_{d\mid n}\frac{d}{n^2+d}.$
I wish to evaluate $\sum\limits_{n\geqslant1}\frac1n\sum\limits_{d\mid n}\frac{d}{n^2+d}.$
Some observations:
Let $f(n)=\sum\limits_{d\mid n}\frac{d}{n^2+d}$.
Then $f(p)=\frac{p^2+p+2}{(p^2+1)(p+1)...
0
votes
1answer
47 views
prove that $ \frac{\sigma(1)}{1}+\frac{\sigma(2)}{2}+\dots+\frac{\sigma(n)}{n} \leq 2 n $
[HMMT 2004] For every positive integer $n$, prove that
$
\frac{\sigma(1)}{1}+\frac{\sigma(2)}{2}+\dots+\frac{\sigma(n)}{n} \leq 2 n
$
If $d$ is a divisor of $i,$ then so is $\frac{i}{d},$ and $\...
2
votes
0answers
46 views
On odd perfect numbers and a GCD - Part III
(Note: This post is an offshoot of this earlier MSE question.)
In what follows, we let $\sigma(x)$ denote the sum of divisors of the positive integer $x$. We also let $D(x)=2x-\sigma(x)$ denote the ...
0
votes
0answers
77 views
Simplify $\sum_{k = 1}^{\left\lfloor{N/2}\right\rfloor} \sum_{i = 1}^{k} \sum_{d_1, d_2 = 1}^{N} \left[{{d}_{1}\, {d}_{2} = i (2\, k - i)}\right]$
This restricted divisor problem is derived from counting the number of reducible quadratics where we have for one of the four main terms
$$\sum_{k = 1}^{\left\lfloor{N/2}\right\rfloor} \sum_{i = 1}^{...
1
vote
1answer
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What is meant by $\sum_{d \le x}f(d)$ in this equation?
Wikipedia's page (here) on the average order of arithmetic functions gives the following as a means of finding such an order using Dirichlet Series:
Define $f$ as an arithmetic function on $n$, and ...
2
votes
1answer
152 views
On variations of a claim due to Kaneko in terms of Lehmer means
In this post (now cross posted as this question on MathOverflow with identificator 362866), for a tuple of positive real numbers $\mathbb{x}=(x_1,x_2,\ldots,x_n)$ we denote its corresponding Lehmer ...
1
vote
2answers
46 views
Stuck with number theory identity about the divisor function
So I have to show that $$\sum_{m=1}^{n} \sigma (m) = \sum_{k=1}^{n} k \cdot \big[\frac{n}{k}\big]$$
where $\sigma$ is the divisor function for x = 1 and [n] is the floor function of n for example [2.4]...
3
votes
1answer
85 views
Can this inequality involving the deficiency and sum of aliquot divisors be improved?
In what follows, we let $n > 1$ be a positive integer. The classical sum of divisors of $n$ is given by $\sigma_1(n)$. Denote the abundancy index of $n$ by $I(n)=\sigma_1(n)/n$.
Denote the ...
0
votes
1answer
53 views
If an arithmetic function $f$ satisfies $f(mn) \leq f(m)f(n)$ (whenever $\gcd(m,n)=1$), is $f$ weakly multiplicative or submultiplicative?
From the preprint On sums of the small divisors of a natural number (Lemma 1, page 2) by Douglas E. Iannucci:
We observe here that the function $a(n)$ is not multiplicative. It is, however, ...
0
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1answer
32 views
Question about a result on odd perfect numbers - Part II
(This question is an offshoot of this earlier one.)
In the paper titled Improving the Chen and Chen result for odd perfect numbers (Lemma 8, page 7), Broughan et al. show that if
$$\frac{\sigma(n^2)}{...
0
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0answers
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Odd perfect numbers having as prime factors exclusively Mersenne primes and Fermat primes: reference request or proposal as an exercise
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 ...
3
votes
1answer
132 views
What is the asymptotic density of positive integers $n$ satisfying $\gcd(n,\sigma(n^2))=\gcd(n^2,\sigma(n^2))$?
This question is an offshoot of this earlier one.
Let $\sigma(x)$ be the sum of divisors of the positive integer $x$. The greatest common divisor of the integers $a$ and $b$ is denoted by $\gcd(a,b)$...
2
votes
1answer
88 views
When does $\gcd(m,\sigma(m^2))$ equal $\gcd(m^2,\sigma(m^2))$? What are the exceptions?
(This question is related to this earlier one.)
Let $\sigma(x)$ be the sum of divisors of the positive integer $x$. The greatest common divisor of the integers $a$ and $b$ is denoted by $\gcd(a,b)$.
...
4
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0answers
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Question about $\varphi(n)+n\mid \sigma(n)$
I want to prove that $$\varphi(n)+n\mid \sigma(n)$$ is impossible , if $\ \omega(n)=2\ $ , in other words , $\ n\ $ has exactly two distinct prime factors.
$\ \varphi(n)\ $ is the totient function ...
1
vote
0answers
44 views
If $q^k n^2$ is an odd perfect number with special prime $q$, then $k \neq 1$ (MSE version)
Note that $\gcd(\sigma(q^k),\sigma(n^2))=i(q)=\gcd(n^2,\sigma(n^2))$ if and only if $\gcd(n,\sigma(n^2))=\gcd(n^2,\sigma(n^2))$. I have therefore undeleted this question, in view of this closely ...
1
vote
0answers
63 views
On odd perfect numbers and a GCD - Part II
Note that $\gcd(\sigma(q^k),\sigma(n^2))=i(q)=\gcd(n^2,\sigma(n^2))$ if and only if $\gcd(n,\sigma(n^2))=\gcd(n^2,\sigma(n^2))$. I have therefore undeleted this question, in view of this closely ...
0
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0answers
35 views
Around a weak form of the Riemann hypothesis inspired in the relationship between the Stolarsky means and the logarithmic mean
Robin's equivalence to the Riemann hypothesis can be written as
$$\frac{\sigma(n)-n}{\gamma+\log\log\log n}<M_{\text{lm}}(\sigma(n),n)\tag{1}$$
for enough large $n$ (it is well-know this suitable ...
10
votes
0answers
215 views
Is there an odd solution of $\varphi(n)+n=\sigma(n)$?
I want to show that the only solution of $$\varphi(n)+n=\sigma(n)$$ for a positive integer $n$ is $n=2$.
What I worked out is that we must have $$\varphi(n)>\frac{n}{2}$$
To show this assume $n$ ...
3
votes
0answers
32 views
Can a squarefree composite number $\ n\ $ satisfy $\ n\mid \sigma(n)+\varphi(n)$?
Let $n$ be a squarefree composite number.
Can $\ \sigma(n)+\varphi(n)\ $ be disivible by $\ n\ $ ?
$\ \varphi(n)\ $ is the totient function and $\ \sigma(n)\ $ the sum of the positive divisors of $...
0
votes
1answer
46 views
The sum of divisor function
In the book of Tom M. Apostol introduction to analytic number theory in the prove of theorem 3.3 the sum of divisor function $d(n)$
$$\sum_{n<x}^{}1 =\sum_{n<x}^{}\sum_{d|n} 1$$
He said that ...
1
vote
1answer
54 views
Is there a number $\mathscr{D}_2 \neq \mathscr{D} = {{3003}^2}\cdot{22021}$ satisfying a certain condition?
(Note: This question is tangentially related to this earlier one.)
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 ...
4
votes
0answers
59 views
On the least number of factors $\sigma(q^{e_q})$ to get the least multiple of $\operatorname{rad}(n)$, on assumption that $n$ is an odd perfect number
I don't know if my question can be answered easily from your reasonings and knowledeges of the theory of odd perfect numbers. I wondered about it yesterday. Now this post is cross-posted on ...
1
vote
0answers
25 views
Can we estimate the divisor function?
I have a question about the estimate of the divisor function.
Let
$$ d(n)=\sum_{d|n}1.$$
I proved that
$$\sum_{n<x}d(n) \ll x \log x \\
\sum_{n<x}\frac{d(n)}{n} \ll (\log x)^2.$$
My question ...
4
votes
2answers
163 views
Why did the Egyptians not represent $2/3$ as a sum of unit fractions in the Rhind papyrus?
The following is taken verbatim from the MathWorld Wolfram page on Egyptian fractions:
An Egyptian fraction is a sum of positive (usually) distinct unit fractions. The famous Rhind papyrus, dated ...
1
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0answers
51 views
Divisibility of odd numbers and its sum of divisors function - Part II
This question is inspired by this earlier one:
Divisibility of odd numbers and its sum of divisors function
In that question, MSE user Juan Moreno claims to have discovered a proof for the following ...
2
votes
1answer
62 views
On bounds for the deficiency of $m^2$, where $p^k m^2$ is an odd perfect number with special prime $p$
Hereinafter, call a number $N$ perfect if $N$ satisfies $\sigma(N)=2N$, where
$$\sigma(x)=\sum_{d \mid x}{d}$$
is the sum of divisors of the positive integer $x$. Denote the abundancy index of $x$ by ...
1
vote
1answer
40 views
An improved inequality for the deficiency function when $\gcd(x,y)=1$, $x > 1$, and $y > 1$
(The following is an attempt to improve on the result contained in this MSE question.)
Let $\sigma(x)$ be the sum of the divisors of a (positive) integer $x$. (For example, $\sigma(2) = 1 + 2 = 3$.)
...
1
vote
2answers
73 views
An inequality for the sum-of-aliquot-divisors function
In what follows, we shall assume that $a$ and $b$ are relatively prime. (That is, $\gcd(a,b)=1$ holds.)
It is known that the inequality
$$\sigma(ab) \leq \sigma(a)\sigma(b)$$
holds for the sum-of-...
17
votes
2answers
307 views
The equation $\sigma(n)=\sigma(n+1)$
In OEIS, the solutions of $$\sigma(n)=\sigma(n+1)$$ where $\sigma(n)$ denotes the sum of the divisors of $n$ including $1$ and $n$ , are shown upto $n=10^{13}$
The entry can be found already by ...
0
votes
0answers
21 views
On certain inequalities involving the sum of divisors function and the main branch of the Lambert $W$ function
In this post we denote the main/principal branch of the Lambert $W$ function as $W(x)$ and the sum of divisors function $\sum_{1\leq d\mid n}d$ as $\sigma(n)$. I add as references the articles from ...
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0answers
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If $p^k m^2$ is an odd perfect number with special prime $p$, then what is wrong about the following factor chain approach to proving $p \neq 5$?
Suppose that $n = p^k m^2$ is an odd perfect number with special prime $p$ satisfying $p \equiv k \equiv 1 \pmod 4$ and $\gcd(p,m)=1$.
That $n$ is perfect essentially means that
$$\sigma(p^k)\sigma(m^...
1
vote
1answer
43 views
Search limit for almost perfect numbers?
The function $$\sigma(n)$$ denotes the sum of the divisors of $n$. Perfect numbers are positive integers $n$ with $$\sigma(n)=2n$$ almost perfect numbers are positive integers $n$ with $$\sigma(n)=2n-...
4
votes
0answers
69 views
Can we decide whether $\sigma(n)-2n=k$ is solvable?
An integer $\ k\ $ is given.
Can we decide whether $\ \sigma(n)-2n=k\ $ has a solution and if yes, can we find one solution efficiently ?
The following even numbers were not "covered" by the range ...
2
votes
1answer
143 views
On characterizations for near-square primes and Fermat primes in terms of equations involving arithmetic functions
In this post we denote the Euler's totient function that counts the number of positive integers $1\leq k\leq n$ such that $\gcd(k,n)=1$ as $\varphi(n)$, and the sum of divisors function $\sum_{1\leq d\...
2
votes
1answer
100 views
From the equation $\sigma(x^{\sigma(y)-1})=\frac{1}{\varphi(x)}(x^{y+1}-1)$ involving arithmetic functions to a characterization of Mersenne exponents
In this post we denote the Euler's totient function that counts the number of positive integers $1\leq k\leq n$ such that $\gcd(k,n)=1$ as $\varphi(n)$, and the sum of divisors function $\sum_{1\leq d\...
1
vote
1answer
76 views
Divisibility of odd numbers and its sum of divisors function
Let us denote as $d(n)$ some proper divisor of $n$ such that $n$ is odd.
I found recently the following
Theorem
If $n=p^\alpha*q$, where $p$ and $q$ are prime numbers, and $q=p^\alpha-\frac{p^\...
0
votes
1answer
39 views
Conjecture on the sum of divisors function
Let us denote as $d(n)$ some proper divisor of $n$ such that $n$ is odd. I am trying to prove that $$\sum_{d\leq\sqrt{n}}d\left(n\right)\nmid n$$
That is, I conjecture that the sum of the proper ...
3
votes
1answer
92 views
From the equation $\sigma(x^{\varphi(y)})=\frac{1}{\varphi(x)}(x^y-1)$ involving arithmetic functions to a characterization of Sophie Germain primes
In this post we denote the Euler's totient function that counts the number of positive integers $1\leq k\leq n$ such that $\gcd(k,n)=1$ as $\varphi(n)$, and the sum of divisors function $\sum_{1\leq d\...
2
votes
1answer
29 views
Ratio of the average of the divisor function $\sigma_k(n)$ for even / odd n
Question:
Is there anything known (for example a proof) about the ratio $\frac{\bar{\sigma_k}(even)}{\bar{\sigma_k}(odd)}$ where $\bar{\sigma_k}$ is the average value which the divisor function ...
2
votes
1answer
129 views
Enrique Santos L's “Proof that no odd perfect number exists”
Background
Let $\sigma(x)$ be the sum of divisors of the positive integer $x$. A number $l$ is called perfect if $\sigma(l)=2l$.
Let $n$ be an odd perfect number given in the so-called Eulerian ...
2
votes
1answer
83 views
On the inequality $m < p^k$ where $p^k m^2$ is an odd perfect number
This question is an offshoot of this earlier one and this other question as well.
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(...
2
votes
1answer
59 views
Conjecture on the solutions to the equation $J(x) = J(x+a)$
Propose $J(x)$, which is a function that takes in a number and outputs the sum of all its factors (including itself)
Firstly, I think it's pretty interesting as it allows you to describe certain ...
1
vote
1answer
48 views
Is it possible to derive $m < p^k$ from the Diophantine equation $m^2 - p^k = 4z$ unconditionally, when it is solvable?
This question is an offshoot of this earlier one.
Allow me to state my question in full:
Is it possible to derive $m < p^k$ from the Diophantine equation $m^2 - p^k = 4z$ unconditionally, where ...
2
votes
1answer
22 views
Convergente of sum of divisors sequence
Let $\sigma$ the application that transforms $n$ into the sum of its divisors (ex : $\sigma\left(6\right)=12$)\
I've proved that
$$
n+1 \leq \sigma\left(n\right) \leq n+n\ln\left(n\right)
$$
I know ...
0
votes
1answer
83 views
sum of the reciprocal of the prime factors of a square free number
Let $n$ an odd square free number, and $p_1, \ldots , p_n$ their distinct prime factors. Ir is true that
$$ \sum\limits_{i=1}^n \frac{1}{p_i} < 1? $$
Otherwise, there exists some conditions to ...
2
votes
1answer
68 views
If $q^k n^2$ is an odd perfect number with special prime $q$, then its index at the prime $q$ is not a square.
Let $N=q^k n^2$ be an odd perfect number with special prime $q$. (That is, $q$ satisfies $q \equiv k \equiv 1 \pmod 4$ and $\gcd(q,n)=1$.)
The index $i(q)$ of $N$ at the prime $q$ is then equal to
$$...
2
votes
1answer
42 views
$(d_{n + 1} - d_n)$ is a geometric progression where $d_1, d_2, \cdots, d_{\tau(m) - 1}, d_{\tau(m)} \mid m$. Prove that $m$ is a prime power.
Given positive divisors $d_1 < d_2 < \cdots < d_{\tau(m) - 1} < d_{\tau(m)}$ of natural $m$ such that the sequence $(d_{n + 1} - d_n)$ is a finite geometric progression $(n \in [1, \tau(m) ...
