For questions on the divisor sum function and its generalizations.

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1answer
27 views

Are there infinitely many $k$ for which $\frac{\sigma(k)}{k}$ is a rational square where $ \sigma(k) $ and $k$ both are square?

Is There some one who can show me if there are infinitely many $k$ for which $$\frac{\sigma(k)}{k}$$ is a rational square where $\sigma(k)$ and $k$ both are square ? Note :$\sigma(k)$ is sum ...
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0answers
28 views

Is this :$\sum_{n=1}^{\infty} \frac{\sigma_{k}(n)}{n!} $ irrational series for every natural number $k$?

Is this: $$\sum_{n=1}^{\infty} \frac{\sigma_{k}(n)}{n!} $$ irrational series for every natural number $k$? Where : $\sigma_{k}(n)=\sigma(\sigma(\sigma(\dots n)))$ is the $k$-th iterate of the sum of ...
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3answers
71 views

Is this : $\lim \sup\frac{\sigma(n)}{n} , n \to\infty $ has a finite limit?

The asymptotic growth rate of the sigma function can be expressed by : $$\lim \sup\frac{\sigma(n)}{n\log(\log(n))}=e^{\gamma}$$ $$n \to\infty$$ according to the above limit , Is this : $$\lim ...
0
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1answer
27 views

How do I show that :$\sigma({p^m})$ is divisible by $4$ if $m=4k+1$ , and $k$ is an integer number?

How do i show this if it's not an open problem :$\sigma({p^m})$ is divisible by $4$ if $m=4k+1$ , and $k$ is an integer number and p is prime number. and $\sigma({p^m})$ is sum divisors of $p^m$ ...
1
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1answer
41 views

When does :$\sigma(\sigma(2n))=\sigma(\sigma(n))$ and $\sigma(n)$ is sum divisors of the positive integer $n$?

Is there someone who can show me When does: $$\sigma(\sigma(2n))=\sigma(\sigma(n))$$ where : $\sigma(n)$ denote the sum divisors of the positive integer $n$ ? Note (1) : I accrossed this problem when ...
0
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1answer
30 views

Sum involving the Möbius function

I have two multiplicative functions $f$ and $g$ and the expression $$\sum_{d\mid n} \mu(d) f(d)g(n/d).$$ In case $f=1$ this is just the Möbius inversion. But what can we say about it in this more ...
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1answer
44 views

How do I show $1$ is not a trivial odd perfect number?

This question related to my this question in MO ,some comments stated that the integer $1$ is trivial answer for this question ,but here i'm very confused when we say that the sum divisors of $1$ is ...
2
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2answers
53 views

Is it possible to determine the number divisors of n! especially for large n?

I read this paper by P. Erdos, page 2. I didn't understand it. How do I determine the number divisors of $n!$ ? I'd like an example application, for example if I want to determine the number divisors ...
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1answer
44 views

Sum of factors of multiplication of different numbers

Given $N$ numbers $n_i$ such that $\forall i \le N, n_i$ $\le10^9$, is there a method to calculate the sum of divisors of their product? For example, given $\{11,15,17\}$ their product would is ...
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2answers
62 views

Is the integer $0$ a deficient number?

It is well known that the divisors of the integer $0$ are all non zero-integers numbers ,the sum of those divisors greater than $0$, then is it a deficient number ? Thank you for any help
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1answer
209 views

How do I prove that there is no other :$k=9,12,18$ for which this fails :$\sigma^k(114) \equiv 0\mod 6 $?

let $\sigma(n)$ be the sum of divisors for a positive integer for example : $$\sigma(6)=1+2+3+6=12$$ . I have performed some calculations in wolfram alpha about the sum divisors of this number: ...
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0answers
41 views

Behaviour of the sum of divisors function via logarithmic means versus an elementary problem equivalent to the Riemann Hypothesis due to Lagarias

It is known the following (see [1], here is an open access in his homepage www.math.lsa.umich.edu/~lagarias/doc/elementaryrh.pdf): Theorem (Lagarias, 2002). Let $\sigma(n)$ denote the sum of the ...
2
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1answer
82 views

$n^2(n-1)\sigma(n)=0 \mod 12$, where $\sigma(n)$ is the sum of divisors function

I would like to ask about the following question: 1) Clarify and complete a proof (by cases) in which the multiplicative function $\sigma(n)$, this is the sum of divisors function, satisfies ...
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2answers
56 views

Solve the equation $\sigma{(n)}=n$

Find all positive $n$ such that $$\sigma{(n)}=n$$ where $\sigma{(n)}$ is the sum-of-divisors function. We write this equation as following: $$\dfrac{\sigma{(n)}}{n}=\sum_{d|n}\dfrac{1}{d}=1$$ ...
4
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1answer
47 views

Can $\sigma(n)-n$ be a proper divisor of $n$?

Let $n\ge 2$ be a natural number, $\sigma(n)$ the sum of its divisors. Can $\sigma(n)-n$ be a PROPER divisor of $n$ ? If $\sigma(n)-n=n$ , $n$ is a perfect number. If $\sigma(n)-n=1$ , $n$ is a ...
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4answers
359 views

Find the sum of reciprocals of divisors given the sum of divisors

Let $d_1, d_2, \cdots d_k$ be all the factors of a positive integer '$n$' including $1$ and $n$. Suppose $d_1 + d_2 + d_3+\cdots+d_k = 72$. Then find the value of $\frac{1}{d_1}+\frac{1}{d_2}+\cdots + ...
0
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1answer
34 views

Probable Candidates for the numbers whose sum of divisors is prime?

What are probable candidates for the numbers whose $\sigma(n)$ (sum of divisors) is prime? I know that the list of probable candidates include perfect squares and odd powers of 2 (specifically only ...
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1answer
36 views

Does :$p(n)=2^{n²+n-1}-n²-n+1 $ abondant for all $n >1$?

let $p(n)=2^{n²+n-1}-n²-n+1 $ , and let $\delta(n)$ be sum of proper divisors of $n\in\mathbb{N}$. After some verifications according to the values of $n>1$ I noticed: $$\delta(p(n))> p(n)$$ ...
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1answer
72 views

Prove that there exist $2015$ consecutive abundant numbers [closed]

A positive integer $N$ is called abundant if the sum of its divisors is greater than $N$: $\delta (N) >N$. My question is: Prove that there exists an integer: $k\in\mathbb N\setminus\{0\}$ ...
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3answers
61 views

For every odd $n\in\mathbb{N}$, is it true that $\sigma(n) < 2n$?

Is the following proposition true? Let $n \in \mathbb{N}$ be an odd number, then $\sigma(n) < 2n$ . For $n=p_1^\alpha p_2^\beta$ it is true : ...
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2answers
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Proof that odd perfect numbers cannot consist of single unique factors?

I'm a high school student, so please point out my mistakes nicely :) So we already know odd perfect numbers cannot be in the form of a square, but how about that they cannot be in this form: ...
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1answer
40 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 ...
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1answer
24 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
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1answer
31 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
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1answer
29 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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2answers
54 views

Is there a solution to this system of equations?

Is there an integer solution to this system of equations? $$\gcd(x, \sigma(x)) = 2x - \sigma(x) = \frac{x}{3} = \frac{\sigma(x)}{5}$$
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0answers
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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 ...
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2answers
96 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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1answer
34 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
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1answer
70 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
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1answer
109 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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3answers
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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
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1answer
41 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
68 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
66 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
41 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
27 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
66 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
82 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 ...
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1answer
39 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
22 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} < ...
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2answers
44 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
31 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
57 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 ...
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1answer
36 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
68 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
62 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
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2answers
454 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
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1answer
32 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 ...