For questions on the divisor sum function and its generalizations.

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What are the applications of Sigma Function?

I read about the Sigma Function today.It tells that- The $\sigma(n)$ is the sum of all the positive divisors of $n$. But I had no idea how they can be useful.What are the practical applications ...
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57 views

How do I make this formula for the primes more concise?

The form I made for the $(n+1)^{th}$ prime $p_{n+1}$ is $\displaystyle1+\sum_{j=1}^{2p_n-1}\lfloor\frac{p_n!^j}{j!}\rfloor-\lfloor\frac{p_n!^j-1}{j!}\rfloor=p_{n+1}.$ Problem is, just like any ...
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1answer
14 views

On the relationship between $\max(p_i)$ and $\omega(b)$, if $\sigma(b^2)/b^2$ is bounded above by a specific number $U$

Let $\omega(x)$ denote the number of distinct prime factors of $x$, and let $\sigma(x)$ be the sum of the divisors of $x$. Denote the abundancy index of $x$ by $I(x) = \sigma(x)/x$. Let the number ...
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1answer
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Question about the validity of a proof involving the abundancy index

Let $\sigma(x)$ be the sum of divisors of $x$, and denote the abundancy index of $x$ by $I(x) = \sigma(x)/x$. Consider the number $y^2 \in \mathbb{N}$, and suppose that I know that $I(y^2) < 4/3$. ...
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1answer
18 views

If $\sigma(N)$ is odd, $N = 2y^2$, and $y$ is not a power of two, does it follow that $\gcd(2,y) = 1$?

Let $\sigma(N)$ denote the sum of the divisors of the number $N$. It is well-known that $$\sigma(N) \equiv 1 \pmod 2 \iff \left\{\{N = x^2\} \lor \{N = 2y^2\}\right\}.$$ Here is my question: If ...
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1answer
147 views

Why can this cosine sum function show all primes less than $N^2$?

I constructed this cosine sum that puts all primes within N on line y=1, and its zeros show the sieve by primes less than N. For $x<N^2$, they are all primes. $$ ...
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1answer
59 views

Are primes less than the sum of divisors?

I am trying to prove that Let $p_n$ be the $n$th prime number, $\sigma (n)=\sum_{d|n}d$. Prove that $$\sigma(n) \le p_n$$ It seems obvious at first glance-to me, at least the sum of divisors of ...
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3answers
46 views

How many numbers less than 100 have the sum of factors as odd?

How many numbers less than 100 have the sum of factors as odd? Answer is 16 This question and explanation is taken from careerbless.com The link given derives the answer using some ...
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3answers
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Proof that $\sum_{d|m} |\mu(d)|=2^n$, where $n$ is the number of distinct prime divisors of $m$?

Given an integer $m$ such that $n$ is denoting the distinct prime divisors of $m$, is there a proof that the sum over the divisors of m of the absolute value of the Mรถbius function $\mu(d)$ is equal ...
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How to generate the sequence of prime building blocks of the colossally abundant numbers: $2, 3, 2, 5, 2, 3, 7, 2, 11, 13, 2, 3, 5, 17, 19, 23, 2,…$

This is the sequence of prime numbers which are the elementary building blocks for the superior highly composite numbers: $2, 3, 2, 5, 2, 3, 7, 2, 11, 13, 2, 3, 5, 17, 19, 2, 23, 7, 29, 3, 31, 2, 37, ...
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1answer
25 views

Does “sum of divisors” function attain every value of the form $kn$?

By the "sum of divisors" function I mean the function $\sigma (n)= \sum_{d|n} d$. If we choose $k=1$ then it is not possible that we have $\sigma (n)=n$ because $n$ always has at least two divisors, ...
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1answer
19 views

On the Density of Deficient Odd Numbers and Abundant Integers

Let $\sigma(x)$ denote the sum of the divisors of $x$. If $\sigma(x) < 2x$, then $x$ is said to be deficient, while if $\sigma(x) > 2x$, $x$ is said to be abundant. (Of course, when $\sigma(x) ...
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On a criterion for almost perfect numbers using the abundancy index

Let $\sigma(x)$ denote the sum of the divisors of $x$, and let $I(x) = \sigma(x)/x$ be the abundancy index of $x$. A number $y$ is said to be almost perfect if $\sigma(y) = 2y - 1$. In a preprint ...
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A cubic relationship for perfect numbers

Combining the identity (the composition of the polynomial $P_3(x)=\frac{x(x+1)(2x+1)}{6}$ with the arithmetical function $N(n)=n$ and the known proof by induction) ...
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On the density of solitary numbers

Let $\sigma(x)$ denote the sum of the divisors of $x$, and let $I(x) = \sigma(x)/x$ be the abundancy index of $x$. If $X$ is the unique solution of $$I(X) = \dfrac{a}{b}$$ (for a given rational ...
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1answer
20 views

Even perfect numbers and a relationship with polygonal numbers

Let $\sigma(m)=\sum_{d\mid m}d$ the sum of divisors function, for example $\sigma(6)=1+2+3+6=12$. Question. I don't know if this exercise was in the literature, and I believe that I know how ...
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16 views

Formula for calculating available quantities

This mathematics novice is having a difficult time with a seemingly easy problem. I'm trying to devise a formula for calculating available quantity for a Widget. Let's say a widget has 10 ...
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Has it been proved that odd perfect numbers cannot be triangular?

(Note: This question has been cross-posted to MO.) Euclid and Euler proved that every even perfect number is of the form $m = \frac{{M_p}\left(M_p + 1\right)}{2}$ where $M_p = 2^p - 1$ is a prime ...
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29 views

Is the following inequality involving the sum-of-divisors and Euler totient functions true?

First Question Is the following inequality involving the sum-of-divisors $\sigma$ and Euler totient $\phi$ functions true? $$\frac{\sigma(N)}{N} \leq \frac{N}{\phi(N)}$$ Second Question When ...
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1answer
48 views

Has limit $\frac{\sigma_0(n)\sigma_2(n)}{(\sigma(n))^2H_n},$ where $H_n$ is the nth harmonic number?

By specialization of an inequality I can write $$2 \sum_{k=1}^{n-1} \frac{1}{d_{k}} \sum_{l=k+1}^{n} \frac{1}{d_{l}}\leq 2\frac{\sigma_0(n)-1}{\sigma_0(n)}\cdot \left( \frac{\sigma(n)}{n} \right)^2, ...
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1answer
33 views

Equation involving 'sigma of integer' function

Let $\sigma (n)$ be the sum of all positive divisors of $n\in\mathbb{N}$. Determine for which least $n$: $$\sigma (x) = n$$ has exactly two and exactly three solutions. The problem also asks for the ...
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1answer
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What is wrong with this proof that $p_1 > \omega(x)$, where $p_1$ is the least prime dividing $x$?

Let $x \in \mathbb{N}$, and let $$x = \prod_{i=1}^{\omega(x)}{{p_i}^{\alpha_i}}$$ be the canonical factorization of $x$. (That is, the $p_i$'s are primes with $p_1 < \ldots < p_{\omega(x)}$.) ...
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1answer
33 views

A computational experiment about identities involving the sum of remainders function

Let $\sigma(m)$ the sum of divisors function and $$S(m)=\sum_{k=1}^m\text{m mod k}$$ the sum of remainders function, then it is know that for integers $m>1$ $$\sigma(m)+S(m)=S(m-1)+2m-1.$$ On the ...
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46 views

Does the Descartes number $D = 198585576189$ have a friend?

Let $\sigma(X)$ be the sum of the divisors of $X$, and denote the abundancy index $\sigma(X)/X$ by $I(X)$. If the equation $I(X) = r/s$ has no solution $X \in \mathbb{N}$, then $r/s$ is said to be an ...
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1answer
31 views

If $\sigma _{1}(n)\mid \sigma _{2}(n)$, does $n$ has to be a perfect square?

Let's say $\sigma _{1}(n)\mid \sigma _{2}(n)$. Can we say, therefore $n$ has to be a perfect square? How to show that?
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1answer
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Exploring the Dirichlet series of the sum of remainder function

I wolud like to learn and understand more some basic facts about Dirichlet series, for wich I want explore the following function, that is called the sum of remainders function, A004125 as Sloane's ...
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1answer
33 views

Writing partition function with divisor function

We know this identitiy. $$P(n)=\frac{1}{n}\sum_{k=0}^{n}\sigma _{1}(n-k)P(k)$$ where $P(n)$ is the partition function and $\sigma _{1}(n)$ is the divisor sum function. Can we pull partition function ...
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Can you discuss when $\sum_{k=1}^n\sum_{m=0}^{k-1}\sec\left(\frac{2\pi m n}{k}\right)$ is defined and when is an integer?

It is know that when we use the trigonometric addition formula for the tangent $$\tan(\alpha+\beta)=\frac{\tan(\alpha)+\tan(\beta)}{1-\tan(\alpha)\tan(\beta)},$$ taking $\alpha=\beta=x$ then from ...
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1answer
30 views

Closed form for $f(n)=\sum_{d\mid n}(-1)^{d}.d$?

Can we find a closed form for $f(n)=\sum_{d\mid n}(-1)^{d}.d$. If $n$ is odd obviously $f(n)=-\sigma(n) $ when $\sigma(n)$ is the sum of divisors function. So when $n$ is even, how to find $f(n)$?
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Is $\sigma(2^r)$ a palindrome (in base $10$) for some $r > 2$, where $\sigma$ is the sum-of-divisors function?

(Note: This post is a bit related to this earlier MSE question.) The title says it all. Is $\sigma(2^r)$ a palindrome (in base $10$) for some $r > 2$, where $\sigma$ is the sum-of-divisors ...
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1answer
41 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 ...
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1answer
15 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 ...
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20 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 ...
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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 $nยฒ$ . My question here ...
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47 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 ...
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1answer
81 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 ...
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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 ...
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44 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 ...
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1answer
67 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 ...
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1answer
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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 $n, m$ : $$\dfrac{\sigma{(n)}}{n}\leq\dfrac{\sigma{(n+m)}}{n+m}$$ ...
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3answers
77 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 ...
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1answer
47 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 ...
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0answers
80 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 ...
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1answer
168 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 ...
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0answers
30 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 ...
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1answer
34 views

Working with divisors [closed]

Compute โˆ… (40), ๐œŽ(124), ๐‘‘(124) and check the equality in ฮฃโˆ…(๐‘‘) = 40. Here's what I've done so far: Not really sure about the summation equality. โˆ… (40) = โˆ… ...
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2answers
36 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 ...
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34 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
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1answer
72 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 ...
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0answers
51 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 = ...