For questions on the divisor sum function and its generalizations.

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2answers
37 views

Number of divisors $d$ of $n^2$ so that $d\nmid n$ and $d>n$

I just wanted to share this nutshell with you guys, it is a little harder in this particular case of the problem: Find the number of divisors $d$ of $a^2=(2^{31}3^{17})^2$ so that $d$ does not ...
1
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1answer
40 views

Is this a good generating function for Sum-of-divisors function?

I have an expression for the sum-of-divisors function defined as $$\sigma(n)=\sum_{d\mid n}d.$$ However I do not know how nontrivial or practical it actually is. Let us define ...
-1
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1answer
31 views

Question on division and remainder

How to solve these type of questions? Which of the following numbers must be added to $5678$ to give a remainder of $35$ when divided by $460$?(Options are) ...
3
votes
1answer
86 views

Solving the number theoretic equation $ \sum_{d|n}{d^4}=n^4+n^3+n^2+n+1 $

I found an interesting problem: Find all $n\in\mathbb N$ such that $$ \sum_{d|n}{d^4}=n^4+n^3+n^2+n+1 $$ If we define $s(n)=\sum_{d|n}{d^4}$, we can show, that $s(mn)=s(m)s(n)$ if $\gcd(m,n)=1$. ...
1
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1answer
43 views

If $\varphi(mn)=\lambda \varphi(m)\varphi(n)$ what should be written for $\lambda$

Respected All. I am studying number theory where I came to know that $\varphi(n), \sigma(n)$ both are multiplicative function ; In other words, if $(m,n)=1$ then \begin{align} ...
0
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1answer
28 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 ...
1
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0answers
33 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
75 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
28 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
43 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
34 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 ...
0
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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
55 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 ...
0
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1answer
46 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 ...
1
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2answers
63 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
4
votes
1answer
214 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: ...
1
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0answers
91 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
89 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 ...
1
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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
votes
1answer
48 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 ...
8
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4answers
370 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 ...
0
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1answer
38 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
76 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: $k\in\mathbb N\setminus\{0\}$ ...
2
votes
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
64 views

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: ...
3
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1answer
41 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
25 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
32 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
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 ...
0
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2answers
55 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
24 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
106 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
64 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
71 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
181 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
43 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
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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
votes
0answers
69 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
71 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
42 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
29 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
27 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
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
83 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
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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 ...
0
votes
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} < ...
1
vote
2answers
48 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
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 ...
1
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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 ...