For questions on the divisor sum function and its generalizations.

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3
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1answer
28 views

If $q^k n^2$ is an odd perfect number with Euler factor $q^k$, can $q = 73$ hold?

Call a number $N$ perfect if $\sigma(N)=2N$ where $\sigma$ is the classical sum-of-divisors function. If $N = q^k n^2$ is an odd perfect number, can $q = 73$ hold? Here is my attempt: Since ...
1
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1answer
18 views

Is it possible to compute $\sigma(AB)$ if $\gcd(A, B) = C > 1$?

Let $\sigma$ be the classical sum-of-divisors function. I know, for one, that $\sigma$ is weakly multiplicative (that is, $\sigma(xy) = \sigma(x)\sigma(y)$ whenever $\gcd(x, y) = 1$). Well, of ...
1
vote
1answer
25 views

A question about $\gcd$ and divisibility

Let $\sigma$ be the classical sum-of-divisors function. Suppose that I have the following equations: $$2n^2 - \sigma(n^2) = \frac{\sigma(n^2)}{q^k}\cdot{\sigma(q^{k-1})}$$ $$2n^2 - \sigma(n^2) = ...
3
votes
1answer
25 views

What is $\gcd(\sigma(q^{k-1}), \sigma(q^k))$?

Let $\sigma$ denote the classical sum-of-divisors function. What is $\gcd(\sigma(q^{k-1}), \sigma(q^k))$? Update: I have transferred the transcript of my attempt to an actual answer to this MSE ...
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votes
2answers
43 views

Is there a solution to this system of equations?

Is there an integer solution to this system of equations? $$\gcd(x, \sigma(x)) = 2x - \sigma(x) = \frac{x}{3} = \frac{\sigma(x)}{5}$$
2
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0answers
16 views

Estimate of sum of divisors up to a certain number?

Let's define $\sigma(n,k) = \sum_{d|n,d\leq k} d$. Notice $\sigma(n,n)=\sigma(n)$, the standard sum of divisor function. It's a standard exercise to show that $$ \sigma(n) \leq n (\ln n + 1) $$ for ...
6
votes
2answers
78 views

When is the sum of divisors a perfect square?

For $n=3$, $\sigma(n)=4$, a perfect square. Calculating further was not yielding positive results. I was wondering is there a way to find all such an $n$, like some algorithm? We know that if ...
1
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0answers
21 views

What is known about multi-perfect numbers?

It is unknown if odd perfect numbers exist and it is known that the even perfect numbers are those of the form $$2^{n-1}(2^n-1)$$, where $2^n-1$ is a (Mersenne-)prime. But what is known about the ...
2
votes
1answer
59 views

Number $N>6$, such that $N-1$ and $N+1$ are primes and $N$ divides the sum of its divisors

The perfect number $6$ is in the middle of the primes $5$ and $7$. It is the only perfect number with this property because odd numbers are not in the middle of two twin primes and even perfect ...
2
votes
1answer
52 views

How to Find the smallest integer with exactly N odd divisors.

Hi All I was trying one problem in which is it asking for the smallest number having N odd divisors. As I know the smallest number having n divisors can be find easily.First we need to find the prime ...
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1answer
15 views

The Sum of Divisors and Abundancy Ratios [closed]

The abundancy ratio, a[n], of a positive integer [n] is the sum of all its positive divisors divided by [n]. Find an integer greater than 1 for which a[n] < 1.001
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3answers
34 views

Basic question about modular arithmetic applied to the divisor sum function $\sigma(n)$ when $n=5p$

While studying the divisor sum function $\sigma(n)$ (as the sum of the divisors of a number) I observed that the following expression seems to be true always (1): $\forall\ n=5p, p\in\Bbb P,\ p\gt ...
3
votes
1answer
38 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
votes
0answers
64 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
votes
1answer
57 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 ...
0
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0answers
35 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
votes
1answer
25 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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vote
2answers
25 views

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 ...
0
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0answers
58 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
votes
1answer
74 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
34 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 ...
0
votes
0answers
19 views

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} < ...
1
vote
2answers
41 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 ...
0
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0answers
27 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 ...
1
vote
0answers
48 views

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
29 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
64 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
58 views

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 ...
28
votes
2answers
426 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 ...
6
votes
1answer
118 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
votes
1answer
60 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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votes
5answers
513 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
59 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
vote
1answer
45 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
34 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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vote
1answer
25 views

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 ...
1
vote
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
55 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 ...
3
votes
0answers
113 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
196 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
43 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
votes
1answer
45 views

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

Is it true $\sigma(A)$/$\sigma(B)$ > = (A/B) ; given B divides A ?
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2answers
262 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 ...
1
vote
1answer
42 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 ...
1
vote
2answers
79 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.
0
votes
0answers
23 views

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
votes
1answer
43 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 ...