For questions on the divisor sum function and its generalizations.

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1answer
35 views

Product Divisors of 2 integers

If $x$ is a positive integer, and $y$ is a positive integer, can the product of the divisors of $x$ equal the product of the divisors of $y$ for some arbitrary $x$ and $y$? (the product of the ...
3
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0answers
62 views

How prove $\sigma(4^p-1)<(2^{p+1}-1)^2$

If $p$ is an odd prime numbers, show that $$\sigma(4^p-1)<(2^{p+1}-1)^2$$ where $\sigma(n)$ stands for the sum of divisors. I thought of using the formula for $\sigma(n)$: If ...
2
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1answer
55 views

Number Theory : Show that $\sigma(n)$ $=$ $2n$ for $n$ $=$ $(2^{m-1})$ $(2^{m} -1)$

I was working through some basic Number Theory Problems when I came across : Given an integer $m$ $≥ 2$ such that $(2^{m} -1)$ is a prime, and $n$ $=$ $(2^{m-1})$$(2^{m} -1)$, then show that ...
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0answers
32 views

On odd perfect numbers $N$ given in the Eulerian form $N = {q^k}{n^2}$, Part II

(Note: This has been cross-posted to MO.) A positive integer $N$ is said to be perfect if $\sigma(N) = 2N$, where $\sigma(x)$ is the sum of the divisors of $x$. An odd perfect number $N$ is said to ...
2
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1answer
23 views

Number of bounded divisors of an integer

Given integers $n,t$, what is an upper bound for the number of integers dividing $n$ in the interval $\{1,\ldots,t\}$? When $t=n$ this boils down to the classical divisor bound ...
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2answers
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Proving associativity of product of two formal sums $\sum_{n = 1}^{\infty} \frac{a_n}{n^x}$

Let $R$ be the set of all formal sums $\sum_{n = 1}^{\infty} \frac{a_n}{n^x}$ where $a_n \in \Bbb{Q}$, where two sums $a, b$ are equal iff $a_i = b_i \ \forall i$. It is indeed a ring with addition ...
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0answers
52 views

On odd perfects and spoofs

This question is an offshoot of this MSE post. Let $\sigma$ be the classical sum-of-divisors function. An odd perfect number $N$ is said to be given in Eulerian form if $\sigma(N)=2N$ and ...
2
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1answer
67 views

On Descartes numbers

This question is an offshoot of this earlier MSE post. Citing Banks, et. al.: "Let us call an integer $n$ a Descartes number if $n$ is odd, and if $n = km$ for two integers $k, m > 1$ such that ...
1
vote
1answer
32 views

For what positive integers is this number-theoretic equation true?

For what odd (positive) integers $x$ is this number-theoretic equation true? $$\gcd(x^2, \sigma(x^2)) = 2x^2 - \sigma(x^2)$$ Here, $\gcd(a, b)$ is the greatest common divisor of $a$ and $b$, and ...
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0answers
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Computational verification request

Let $I(x) = \sigma(x)/x$ be the abundancy index of $x$. Note that $\sigma$ is the classical sum-of-divisors function. Previously, I computed for $u$ in the inequality $$\sqrt{3} < ...
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2answers
38 views

Infinite number of pairs of distinct integers $(m,n)$ such that $\sigma(m^2)=\sigma(n^2)$

I have to prove that there exist infinite number of pairs of distinct integers $(m,n)$ such that $\sigma(m^2)=\sigma(n^2)$ ($\sigma$ is sum of divisors). I tried solving it by using the fact that ...
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0answers
25 views

Another question related to the sum-of-divisors function

If ${q^k}{n^2}$ is an odd perfect number where $q$ is prime with $q \equiv 1 \pmod 4$ and $\gcd(q,n)=1$, then $\sigma(q^k)/n \neq \sigma(n)/q^k$, where $\sigma$ is the classical sum-of-divisors ...
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0answers
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A question related to the sum-of-divisors function

In what follows, let $\sigma$ be the sum-of-divisors function, and assume that we have $\sigma(a^b)\sigma(c^2)=2{a^b}{c^2}$ together with $\gcd(a,c)=1$ and integers $a, c > 1$. Let ...
0
votes
1answer
28 views

Application: Sum of Digits

if a five digit number N is such that sum of its digit is 29, can N be square of an integer? Suppose N be abcde, where a+b+c+d+e = 29. Can square of any number less than abcde is equal to abcde ...
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0answers
63 views

Maybe Fermat primes are infinite?

A Fermat prime is a prime $p$ in the form $2^{2^n}+1$, for some integer $n\ge 0$. It is actually unknown if there infinitely many such primes. Despite the title, here we propose an argument in its ...
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0answers
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A question on the truth of a conjectured biconditional

(Note: This has been cross-posted to MO.) Suppose I have the following series of implications (the underlying assumption is $I(q^k)I(n^2) = 2$), where $I(x) = \sigma(x)/x$ is the abundancy index of ...
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2answers
423 views

A number $n$ which is the sum of all numbers $k$ with $\sigma(k)=n$?

For a positive integer $n$, let us define a set $$A_n = \{ k\in\mathbb{N} \mid \sigma(k) = n \}$$ where $\sigma$ is the divisor-sum function (a well-known multiplicative number-theoretic function). ...
1
vote
1answer
31 views

On Inequality Concerning Deficient Numbers

By Definition a positive integer $N$ is d-deficient if $\sigma(N)=2N-d$. Am I correct if I say that the inequality $N>d$ always hold for this definition? Here is my attempt to show that it is ...
5
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1answer
90 views

The number of divisors of a number whose sum of divisors is a perfect square

Let $n$ denote a non-prime whose sum of divisors is a perfect square. I have noticed a few surprising facts on the number of divisors of $n$: It is either prime or semi-prime or $27$ in all cases ...
3
votes
1answer
53 views

What proportion of the positive integers satisfy this number-theoretic inequality?

Let $\sigma(x)$ be the sum of the divisors of the positive integer $x$, and let the abundancy index of $x$ be defined as $$I(x) = \frac{\sigma(x)}{x}.$$ My question is this: What proportion of the ...
0
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1answer
48 views

John and Gary are playing a game. John spins a spinner numbered with integers from 1 to 20.

John and Gary are playing a game. John spins a spinner numbered with integers from 1 to 20. Gary then writes a list of all of the positive factors of the number spun except for the number itself. Gary ...
3
votes
1answer
60 views

A simple lemma on divisors…

Let $D$ be a strictly positive divisor defined on a compact Riemann Surface such that $\operatorname{dim} \mathfrak{L}(D)=1+\operatorname{deg} D$. There exists a point $p \in X$ such that ...
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5answers
478 views

Identity with nested sum taken over divisors of $\gcd$'s

For computational reasons, I want to show that the following holds true: Let $n_1,n_2,N\in \mathbb{N}$. One has $$\Large \sum_{a\mid \gcd(n_1,N)}\sum_{b\mid \gcd(n_2,\frac{n_1N}{a^2})} ab ...
2
votes
1answer
57 views

Improving the inequality $x\sigma_1(x) \leq \sigma_1(x^2)$ for $x \in \mathbb{N}$

Let $\mathbb{N}$ be the set of positive integers. For $x \in \mathbb{N}$, $\sigma_1(x)$ gives the sum of the divisors of $x$. (For example, $\sigma_1(3) = 1 + 3 = 4$.) We call the ratio $I(x) = ...
1
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1answer
44 views

Prove no odd number can be abundant.

A perfect number is a number for which the sum of its proper divisors is exactly equal to the number. For example, the sum of the proper divisors of $28$ would be $1 + 2 + 4 + 7 + 14 = 28$, which ...
2
votes
1answer
32 views

How to calculate $\sigma(x^k)$?

If $n = 2^k$, then $\sigma(n) = 2 \times 2^k - 1$, For example: $$\sigma(28) = \sigma(2^2) \times \sigma(7) = 7 \times 8 = 56$$ But how about: $$\sigma(100) = \sigma(2^2) \times \sigma(5^2) = 7 ...
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1answer
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Question on Sum of Divisor?

I know $\sigma(m)=24$ for $m=\{14,15,23\}$ but how can we find this numbers? Here is what I did Let the prime factorization of $m$ be $$m=p_1 ^{\alpha _{1}}p_2 ^{\alpha _{2}}\cdot\cdot\cdot p_k ...
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1answer
34 views

Determine if $\sum_{q=1}^{\lceil n/2\rceil}R_q(n)$ gives the number of divisors of $n$.

Let $$R_q(n)=\left\{\begin{array}{lll} r\left(\dfrac n{2q-1}\right)&\text{if }(2q-1)\mid n\\ 0&\text{otherwise}\end{array}\right\},$$ where $r(n)$ is the ruler function, i.e., the $2$-adic ...
2
votes
1answer
54 views

Prove a property of the divisor function (Part 2)

Further to this MSE question, I would like to pose a follow-up inquiry: If $n \in \mathbb{N}$ and $(\sigma(n) - n) \mid (n - 1)$, does it follow that $n$ and $\sigma(n)$ would have to be coprime, so ...
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0answers
111 views

Divisor summatory function for squares plus one

As an exercise for my Analytic Number Theory course, I need to prove, using Dirichlet hyperbola method, that: $\sum_{n\leq x}\tau(n² + 1)= {3\over\pi}x\log(x) + O( x)$, where $\tau(n)=\sum_{d|n}1$ ...
2
votes
1answer
192 views

Proof regarding Robin's inequality (RI).

Let $\sigma$ be the divisor sum function, $\gamma$ the Euler-Mascheroni constant and $n>5040$. Robin showed that if the inequality$$\displaystyle \sigma(n)<e^{\gamma}n\log\log n$$ ever fails, it ...
0
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1answer
38 views

Function $f(n) = 2^{\omega(n)}\mu^2(n)$

Let $$f(n) = 2^{\omega(n)}\mu^2(n)$$, where $\omega(n)$ is number of distinct prime divisors of $n$ and $\mu(n)$ is Moebius function. I want to simplify it. As long as $$ ...
0
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1answer
41 views

About the sigma function and an interesting inequality. [closed]

Is it true $\sigma(A)$/$\sigma(B)$ > = (A/B) ; given B divides A ?
5
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1answer
189 views

What's wrong in this proof of $10$ is a solitary number?

Friendly numbers are two or more natural numbers with a common abundancy, the ratio between the sum of divisors of a number and the number itself. Two numbers with the same abundancy form a friendly ...
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vote
1answer
41 views

how to sum the divisors of N mod K if all I have is N mod K?

The input to this problem is N. I have to calculate 2 things: 1 - N! mod (10^9 + 7) 2 - sum of all divisors of N! mod (10^9 + 7) I know how to do the first step, I'm wondering if there is a way ...
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2answers
77 views

About the 'sigma' function.

Is it true that if $n$ divides $m$ , $\sigma(\frac mn) \leq \frac{\sigma(m)}n$. If so this has a bearing on counterexamples to Robin's inequality.
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0answers
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Sum of convolution of divisor function [duplicate]

For every integer $k$ let $d_k: \mathbb{N} \rightarrow \mathbb{C}$ be defined recursively as $d_0 = \mathbf{1}$, $d_k = d_{k-1} * \mathbf{1}$. So for example $d_1 (n) = d (n) = \sum_{d \vert n} 1$ is ...
0
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1answer
42 views

Why is this Moebius equivalence true?

I would like to know why the following is true: $$\tau(n^2) = \sum_{d | n} \mu(n/d)(\tau(d))^2$$ I cannot derive it. It is on OEIS but I'd like to know how this was found. $\tau(n)$ is the count of ...
2
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0answers
47 views

Number of divisiors of $n$ less than $m$

I'm looking for a closed- or alternative-form of the function that counts the number of divisors of an integer $n$ that are less than some integer $m$ (interested in $m < n$, obviously): $ ...
0
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1answer
87 views

Who should I ask for Robin's paper? At any rate, I want to find out if a similar result to his can be achieved with 36 instead of 12.

Robin proved unconditionally that for $\ n \ge 3$ , $$ \sigma(n)<\left(e^\gamma+{\log\log12\left({\frac73}-e^\gamma \log\log12\right)\over (\log \log n)^2}\right)n \log \log n. $$ I need a similar ...
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0answers
57 views

If Lagarias' inequality is wrong, are there infinitely many counterexamples to it?

I do know that since Robin's (RI) and Lagarias' (LI) inequalities are both equivalent to RH, they're also equivalent one another, hence if RI is false, so is LI. And Robin proved there are infinitely ...
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0answers
30 views

Could you give a definition of what is a superior highly composite number using only words?

I know very well what is a superior highly composite number, but I would like to see how could we (roughly) define what is a superior highly composite number using only words (using no equations and ...
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0answers
64 views

With odd $\ n>9, \sigma(n) < {11\over 16} e^{\gamma} n \log \log n $?

If that's not the case, do we know anyway some upper bound better than that given by Robin's inequality, since it has been shown that it holds for all odd numbers > 9 ( Choie, YoungJu, et al. "On ...
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0answers
88 views

Find a closed form for the constant term

In a previous question, an asymptotic expansion was provided for the weighted divisor summatory function $\displaystyle \frac {d(n)}{n}$: $$\sum_{n\leq ...
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1answer
69 views

Why are multiples of abundant numbers also abundant numbers?

I am currently working on Project Euler Problem 23 which involves abundant numbers. In short, abundant numbers are numbers that are less than the sum of their proper divisors. For example, 12 has ...
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0answers
63 views

What are the patterns in the number of divisors $d(n)$ of the highly composite numbers?

I am trying to understand the patterns in the number of divisors $d(n)$ of the highly composite numbers. The numbers marked with an asterisk are the superior highly composite numbers. The first ...
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0answers
84 views

Given an integer $n$ find smallest integer $i$ such that $σ(i)=n$. Smallest Inverse Sum of Divisors

Hi All I need some help I am trying to solve this problem which involves computation of sum of divisors and its inverse. In other words Given an integer $n$ find smallest integer $i$ such that ...
3
votes
2answers
77 views

Infinite series involving sum of divisors function

Is there much known about infinite series of the form $\sum_{n=1}^{\infty}\frac{\sigma_{1}(n)}{n}q^{n}$ where $\sigma_{1}(n)$ is the sum of divisors function. I am particularly interested in ...
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1answer
92 views

the average order of divisor function

In Analytic number theory by Apostol there's a theorem: $$\sum_{n\le x} \sigma(n)= \frac{1}{2} \zeta(2)x^2 + O(x\log x)$$ and then it claims that because we know that $\zeta (2)= \frac{\pi^2}{6} $ ...
1
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1answer
38 views

Prove that for any $n$ having at least two distinct prime factors, there exists a perfect square between $n$ and $\sigma(n)$

I came up with a new problem, described as follows: Prove that for any $n$ having at least two distinct prime factors, there exists at least one perfect square between $n$ and $\sigma(n)$ I will post ...