For questions on the divisor sum function and its generalizations.

learn more… | top users | synonyms

0
votes
0answers
17 views

Odd perfect numbers and $\sum_{\substack{D\mid 2n,D<\sqrt{2n}}}(D+\frac{2n}{D})$

It is known that if $m>1$ isn't a perfect square integer (isn't a square number) then the sum of divisor function can be written as $$\sigma(m)=\sum_{\substack{d\mid ...
3
votes
1answer
33 views

Can you prove this identity involving the divisor sum function?

Let $$\sigma(n)=\sum_{d\mid n}d$$ the sum of divisor function. I've deduced, but I don't know how prove directly $$\sigma(n)=\sum_{m=1}^{n}\sum_{k=1}^{m}(-1)^{nk}\cos\left(\frac{\pi ...
0
votes
1answer
10 views

If $N = {q^k}{n^2}$ is an odd perfect number given in Eulerian form, does $\sigma(n^2)/q$ divide $2n^2 - \sigma(n^2)$?

Let $\sigma(x)$ be the sum of the divisors of $x$. A number $X$ is called perfect if $\sigma(X) = 2X$. Denote the abundancy index $\sigma(X)/X$ by $I(X)$. If $N$ is odd and perfect, then $N$ can be ...
0
votes
0answers
16 views

Can you analyze this identity involving the sum of divisors function and $rad(n)=\prod_{p\mid n}p$?

Let $\sigma(n)$ the sum of divisors function, then by Mobius inversion formula $$\sigma(n)=n-\sum_{\substack{d\mid n,d<n}}\sigma(d)\mu\left(\frac{n}{d}\right),$$ and since this function is ...
2
votes
0answers
22 views

How do i show this :$\lim_{k\to\infty} \frac{\sigma_{2k+1}(n)}{\sigma_{2k-1}(n)}=n²$ if it is true?

I run some computation in wolfram alpha I find for many fixed values of $n$ and for an arbitrary integer $k$ the ratio : $\frac{\sigma_{2k+1}(n)}{\sigma_{2k-1}(n)}$ close to $nĀ²$ . My question here ...
0
votes
0answers
39 views

On even almost perfect numbers other than the powers of two, as compared to odd perfect numbers given in Eulerian form

(Note: I have edited this question to conform to the further details added in the cross-post to MO.) Let $\sigma(x)$ be the sum of the divisors of $x$. We say that $X$ is almost perfect if ...
0
votes
1answer
80 views

Any counter example for this claim?

I would like to proof or disproof this claim ,but i don't have enough information about divisor function structure . Claim : for any positive integer $x, y ,n $ such that :$x\neq y$ and ...
3
votes
0answers
40 views

A number $n$ has $12$ divisors and $d_{d_4-1} = (d_1+d_2+d_4)d_8$.

Find a number $n$ which has - $1.$ $12$ divisors $(1 = d_1 < d_2 < \cdots <d_{12}=n )$ and $2.$ $d_{d_4-1}=(d_1+d_2+d_4)d_8$. Note: This is a problem from Russian Mathematical Olympiad ...
-1
votes
0answers
17 views

Is there a closed form of $n$ for :$h(n)=\frac{\sigma(n)}{n}$ for which $n$ is coprime to $\sigma(n)$?

It is well known that $h(n)=\frac{\sigma(n)}{n}$ is quit irrigulrar,I'm very interesting to know more about it's behavior and i would like to know more about coprimality characteristic then it must be ...
0
votes
0answers
43 views

Inequality between two sums of numbers of divisors

Let $D_b(m)$ be the number of divisors of $m$ that are less than $b$. Neil Sloane has suggested that the number of binary quadratic forms $Ax^2 + Bxy + Cy^2$ with integral coefficients, discriminant ...
3
votes
1answer
56 views

Is $\lim\limits_{m\to\infty}\frac{1}{m}\sum_{n=1}^m \cos\left(\frac{2\pi nj}{s}\right)$ “useful”?

In this answer it is explained that a big reason that the nth prime formula discussed with (13) and (14) isn't "useful" is because $\lfloor\frac{x}{b}\rfloor-\lfloor\frac{x-1}{b}\rfloor$ doesn't have ...
0
votes
1answer
20 views

for which conditions of postive integer $n, m >0$ :$\dfrac{\sigma{(n)}}{n}\leq\dfrac{\sigma{(n+m)}}{n+m}$ hold?

I would like to know more about behavior of growth rate of sum divisor function I accross this problem then :for which conditions for $n, m$ : $$\dfrac{\sigma{(n)}}{n}\leq\dfrac{\sigma{(n+m)}}{n+m}$$ ...
3
votes
3answers
75 views

An identity involving $[\sigma(n)]^2$

For a positive integer $n$, let $\sigma(n)$ denote the sum of the divisors of $n$. For example, $\sigma(1)=1$, $\sigma(2)=3$, $\sigma(4)=7$, etc. I would like to prove the following identity: For ...
2
votes
1answer
43 views

What inequalities similar Lagarias' statement are easy to prove?

Let $$H_n=1+\frac{1}{2}+\cdots+\frac{1}{n},$$ the nth harmonic number and $$\sigma(n)=\sum_{d\mid n}d,$$ the sum of divisor function, for example $\sigma(6)=12$. I believe that this could be a nice ...
0
votes
0answers
79 views

If $N = {q^k}{n^2}$ is an odd perfect number given in Eulerian form, is $n$ a square?

(Note: There is a related question, already with an answer, in MO. Additionally, this question has been cross-posted to MO.) Let $x$ be a positive integer, and let $\sigma(x)$ denote the sum of the ...
6
votes
1answer
160 views

A conjecture about prime numbers based on $\sigma_1(n)$ and the Highly Abundant Numbers

I am trying to find the smallest expression $E(n)$, whose distances between the value of the expression and the next prime closer to the expression, $\mathcal{N}(E(n))$, and from the expression to the ...
1
vote
0answers
29 views

If $N = {q^k}{n^2}$ is an odd perfect number given in Eulerian form, is $n$ squarefree?

(Note: This question has been cross-posted to MO.) Let $x$ be a positive integer, and let $\sigma(x)$ denote the sum of the divisors of $x$. So for example, $\sigma(6) = 1 + 2 + 3 + 6 = 12 = 2 ...
1
vote
1answer
34 views

Working with divisors [closed]

Compute āˆ… (40), šœŽ(124), š‘‘(124) and check the equality in Ī£āˆ…(š‘‘) = 40. Here's what I've done so far: Not really sure about the summation equality. āˆ… (40) = āˆ… ...
1
vote
2answers
33 views

What is an upper bound for $\frac{4x - \sigma(x)}{3x - \sigma(x)}$ when $x$ is deficient?

Let $x$ be a positive integer, and let $\sigma(x)$ be the sum of the divisors of $x$. For example, $$\sigma(12) = 1 + 2 + 3 + 4 + 6 + 12 = 28.$$ I would like an upper bound for the expression ...
1
vote
0answers
30 views

Fourier series concerning Gibbs constant and the divisor function.

It is quite a remarkable function I found. It seems, though, that I may be staring at something trivial, which is hopefully not the case. I would like some opinions. The function is ...
4
votes
1answer
64 views

Fourier transform for sum-of-divisors function

I found what seems to be a Fourier transform formula for the sum-of-divisors function. $$\sigma(x)=\sum_{d|x}d$$ The "trick" is that it requires a limit. I don't know if this matters or not but ...
2
votes
0answers
49 views

Reference request: Do any papers on odd perfect numbers approach the problem using the following equation?

(Note: This question has been cross-posted to MO.) Do any papers on odd perfect numbers approach the problem using the following equation? $$N - (q^k + n^2) + 1 = ...
1
vote
0answers
35 views

If $\gcd(Z,\sigma(Z))=1$ and $N=Z\sigma(Z)$, is $N$ always friendly?

This question is a generalization / offshoot of this earlier MSE post: If $\gcd(Z,\sigma(Z))=1$ and $1<N=Z\sigma(Z)$, is $N$ always friendly? Here, $\gcd(a,b)$ is the greatest common divisor ...
1
vote
0answers
79 views

Why is proving that $10$ is solitary considered very difficult?

The title says it all. We denote the sum of the divisors of $x$ by $\sigma(x)$. The ratio $I(x)=\sigma(x)/x$ is called the abundancy index of $x$. If $I(m)=I(n)$, then $\{m,n\}$ is called a ...
1
vote
0answers
41 views

Is $p(p + 1)$ always a friendly number for $p$ a prime number?

Let $\sigma(x)$ denote the sum of the divisors of $x$. We call the ratio $I(x) = \sigma(x)/x$ as the abundancy index of $x$. A positive integer $N$ is friendly if there exists another positive ...
0
votes
1answer
36 views

If $\sigma(N) = aN + b$, where $\gcd(a, b) = 1$, does it follow that at least one of $N$'s factors is solitary?

Let $\sigma(x)$ be the sum of the divisors of $x$. If $\sigma(N) = aN + b$, where $\gcd(a, b) = 1$, $a \geq 2$, and $b$ could be negative, does it follow that at least one of $N$'s factors is ...
1
vote
1answer
53 views

If $n$ satisfies $\left(-3+\sqrt{1+8n}\right)\sigma(n)=4\left(-1+\sqrt{1+8n}\right)\phi(n)$ then is an even perfect number?

Let an integer $m\geq 1$, and $\sigma(m)$ is the sum of positive divisors function, and $\phi(m)$ is Euler's totient function, counting the number of integers $1\leq k\leq m$ such that $gcd(k,m)=1$ ...
2
votes
0answers
42 views

With $rad(N)=\prod_{p|N}p$, if $N$ is even and $\frac{2+rad(N)}{8}\left(\sum_{\substack{d|N,d<rad(N)}}d\right)=N$ then is perfect?

In the literature (see for example sites and paper concerning to the abc conjecture, I say this as reference and by caution to avoid mistakes) is defined the arithmetical function $rad(n)$ as $1$ if ...
2
votes
1answer
34 views

Iterated $\sigma^k(n)$, excluding the possibility that primes $p|\sigma^t(n)$, $t< k$ divides an odd perfect number $n$

Let $m\geq 1$ an integer and $\sigma(m)=\sum_{d|m}$ the sum of positive divisors function. A positive integer is said to be perfect if and only if $\sigma(n)=2n$. We have that $\sigma(m)$ is a ...
1
vote
0answers
29 views

what is number of unique positive divisor of an integer n?

I am working on my number theory assignment and we have this theory that states that sum of Euler's totient function of all positive divisor equals the number itself this is just a little back ...
1
vote
0answers
48 views

How to simplify a sum of complex divisors?

This question arises from Project Euler 153. That problem asks for the sum of all complex divisors of all natural numbers up to a maximum, where a complex divisor is a complex number of the form a + ...
0
votes
0answers
24 views

Criteria for almost perfect and $p$-deficient numbers

Let $\sigma$ be the (classical) sum-of-divisors function. A number $n$ is called almost perfect if $\sigma(n) = 2n - 1$. If $\sigma(m) = 2m - p$ for some integer $p > 1$, then $m$ is called ...
0
votes
0answers
35 views

Sum of Divisors Multiplicative

I wish to prove that the sum of the divisors function, $\sigma(x)$, is multiplicative, i.e. $$\sigma(m\cdot n)=\sigma(m)\cdot\sigma(n)$$ if $gcd(m,n)=1$. I start by claiming that since $m$ and $n$ ...
0
votes
0answers
13 views

When does the system $\sigma(n) = aq$, $\sigma(q^k) = an$ have any solutions for prime $q$, $\gcd(q, n) = 1$, $k > 1$, and $k \equiv 1 \pmod 4$?

This question is an offshoot of this earlier MSE post. When does the system $$\sigma(n) = aq$$ $$\sigma(q^k) = an$$ have any solutions for prime $q$, $\gcd(q, n) = 1$, $k > 1$, and $k \equiv 1 ...
0
votes
2answers
29 views

Does this equation have any solutions for prime $q$ and $k > 1$, $k \equiv 1 \pmod 4$?

Does this equation have any solutions for prime $q$ and $k > 1$, $k \equiv 1 \pmod 4$? $$\sigma\left(\frac{1}{2}\sigma(q^k)\right) = 2q$$ Here $\sigma$ is the classical sum-of-divisors function. ...
0
votes
2answers
32 views

Find infinitely many n such that $\sigma (n) \le \sigma (n-1)$

This number theory problem has me confused. It is finding infinitely many n such that $\sigma (n) \le \sigma (n-1)$. I know that n should be prime by the examples I have tried, but I'm not sure where ...
4
votes
1answer
80 views

On the binomial series $(1+\frac{1}{8n})^{1/2}$, where $n$ is an even perfect number

Since $\sqrt{1+8n}=\sqrt{8n}\sqrt{1+\frac{1}{8n}}$, and $\frac{1}{8n}<1$ when $n>1$ is an integer, then we can express the real number $\sqrt{1+\frac{1}{8n}}$ by its binomial series. This series ...
1
vote
1answer
43 views

Divisor Function Proofs

I am trying to decipher the proofs to the following statements: Let $d$ be the divisor function, then; show that $d(n)$ is odd if and only if $n$ is a square Show that for a given $n\geq 2$, there ...
6
votes
0answers
118 views

Limit superior, limit inferior and a series involging $\sum_{k\nmid n}$k, where $1\leq k\leq n$

The purpose of this post is state assertions by the use of statements and hypothesis in an expository way and after I am asking for reasonable unconditionally results that you can provide us. Using ...
0
votes
0answers
45 views

Does the following equation have any solutions in $\mathbb{N}$?

Let $\mathbb{N}$ be the set of positive integers. The function $\sigma(N)$ gives the sum of the divisors of $N$. My question is: Does the following equation have any solutions for $x \in ...
2
votes
0answers
48 views

Is the odd part of even almost perfect numbers (other than the powers of two) not almost perfect?

Let $\sigma(x)$ denote the sum of the divisors of $x$. A number $M$ is called almost perfect if $\sigma(M) = 2M - 1$. If $M$ is an even almost perfect number, then the only known examples for $M$ ...
1
vote
0answers
52 views

sum of digits = sum of factors

Assume that some set $A$ is defined as $A=\{x|x\in Z \ni S(10,x)=\sigma_1(x)\}$ where $S(10,x)$ returns the sum of all of x's digits in base 10, and $\sigma_1(x)$ returns the som of all prime factors ...
1
vote
0answers
31 views

Is $\frac{4x^2 - 1}{3x^2}$ an abundancy outlaw?

Let $\sigma(x)$ denote the sum of the divisors of $x$. For example, $\sigma(6) = 1 + 2 + 3 + 6 = 12$. We call the ratio $I(x) = \sigma(x)/x$ the abundancy index of $x$. A number $y$ which fails to ...
3
votes
2answers
103 views

Improvement on $\phi(n)\sigma(n)/n^2$ bounds?

We have: $$\dfrac{6}{\pi^2}\lt\dfrac{\phi(n)\sigma(n)}{n^2}\le1$$ with equality iff $n=1$. Wikipedia Are there any known improvements on these bounds? APPENDUM For $n$ prime, ...
2
votes
1answer
58 views

Does OEIS sequence A059046 contain any odd squares $u^2$, with $\omega(u) \geq 2$?

Does OEIS sequence A059046 contain any odd squares $u^2$, with $\omega(u) \geq 2$ (where $\omega(x)$ is the number of distinct prime factors of $x$)? Here are the first sixty-two terms: A059046 - ...
1
vote
0answers
15 views

How to express this sum of certain non-divisors in a different form?

I need help finding a way to express this sum. $$\sum_{\substack{b\space\nmid\space x\\{\left\lfloor{\frac{x}{b}}\right\rfloor}\space\text{is even}}}^xb$$ I need to express this in a different form ...
0
votes
2answers
29 views

Finding a specific number whose divisors exceed a fixed value.

I am trying to find smallest number s.t. it's square has more than 8 million divisors. It might seem impossible for hand and even while solving with a computer, it requires great amount of time (but I ...
0
votes
0answers
23 views

A follow-up number-theory question on the deficiency function $D(x) = 2x - \sigma(x)$

This question is a follow-up to these previous posts: MSE1 and MSE2. Let $x, y$ be positive integers. We call $\sigma(x)$ the sum of the divisors of $x$. Let the deficiency function $D(x)$ be ...
4
votes
0answers
108 views

A mixture with ingredients of two equivalences with Riemann Hypothesis

Let $f(x)=x\cdot(\log x)^x$ for $x\geq 2$, then integrating $\log f(x)=\int_2^x f'(t)/f(t)dt$, it is easy to prove the first statement of following, and directly if we put $x=e^H_n$ and add $H_n$ the ...
1
vote
0answers
27 views

Over what subset of $\mathbb{N}$ is the deficiency $D(x) = 2x - \sigma(x)$ a weakly multiplicative function?

This is an offshoot of this MSE question which was posted earlier today. Let $\mathbb{N}$ be the set of natural numbers (i.e., positive integers). We call $\sigma(x)$ the sum of the divisors of $x$. ...