For questions on the divisor sum function and its generalizations.

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Can this heuristic about Sorli's conjecture and odd perfect numbers be made rigorous?

Let $N = q^k n^2$ be an odd perfect number with Euler prime $q$. That is, we have $q \equiv k \equiv 1 \pmod 4$. Sorli (page 89) conjectured that $k=1$ always holds. Suppose we rewrite $N$ as $$N = ...
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What are the mathematical consequences if $10$ is proved to be solitary?

Let $\sigma(x) = \sigma_{1}(x)$ denote the sum of the divisors of $x$, and let $$I(x) = \dfrac{\sigma(x)}{x}$$ be the abundancy index of $x$. For example, $$\sigma(10) = 1 + 2 + 5 + 10 = 18$$ so that ...
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Divisor Function over a Quadratic

The divisor function is defined as $\sigma_1(n)=\sigma(n)=\sum_{d\mid n}d$. Consider the divisor function over a quadratic $$f(x)=\sigma(a x^2+bx+c)$$ Where $a,b,c \in \mathbb{Z}$ (note we allow $a, b$...
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Project Euler's, Problem #565

Project Euler's, Problem #565 states: Let $\sigma(n)$ be the sum of the divisors of $n$. E.g. the divisors of $4$ are $1, 2$ and $4$, so $\sigma(4)=7$. The numbers $n$ not exceeding $20$ ...
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Justify $\lim_{n\to\infty}n^p\int_0^1\sum_{k=n}^\infty\frac{\sigma(k)e^{x/k}}{k^{p+2}\log\log k} dx=\frac{e^\gamma\int_0^1f(x)dx}{p}$

Inspired in PROBLEM 207, La Gaceta de la Real Sociedad Matemática Española, Vol. 16, N0. 3 (page 507 in spanish, proposed and solved by Furdui), I've tried write examples of this new statement ...
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Explaining an integral involving the divisor function

In a 1973 paper by Martinet, Deshouilliers and Cohen, $A(x)$ is defined as $$A(x)=\lim_{N\to\infty}\frac{\#\{n\leq N\mid \frac{\sigma(n)}{n}≥x \}}{N}$$ where $\sigma(n)$ is the "sum-of-divisors" ...
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Exploring congruences and identities involving Mersenne primes and the terms of Lucas-Lehmer test

When I was exploring congruences$\dagger$ involving the terms $S_k$ defined in Lucas-Lehmer test (this reference is the Wikipedia, but the main reference is Crandall and Pomerance, Prime Numbers: A ...
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More claims deduced from even perfect numbers, and what is the meaning of such calculations

I show the following examples of easy deductions for even perfect numbers $n=2^{p-1}\cdot(2^p-1)$, where $b=2^p-1$ is its associated Mersenne prime, and after I am asking to you about the $\Leftarrow$ ...
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1answer
52 views

Sum of divisors and indices: $\sigma (n)= 2^k$

Are there infinitely many positive integers (say $n$ is one of them), sum of whose divisors are powers of two, i.e, $ \sigma (n)= 2^k $ ?
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Explaining a Series Expansion for the Divisor Function

The "sum-of-divisors" function, defined as $\sigma(n)=\sum_{d\,\mid\,n}d$, can be expressed as $$\sigma(n)=\sum_{i=1}^n\sum_{j=1}^i\cos\Big(2\pi n \frac{j}{i}\Big)$$This expansion makes logical sense. ...
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3answers
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Elementary number theory sum of divisors

Let the sum of the divisors of a number $N$ be equal to $s$(excluding N itself) then show that if $s=N$ then show that N is a perfect number. I tried to use the basic formula for sum of divisors but ...
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1answer
56 views

Does satisfy $f(n)=\frac{\sigma(n)}{n^2}$ the hypothesis of Halasz’s inequality?

Let $\sigma(n)=\sum_{d\mid n}d$ the sum of divisor function. I would like to know if I can write an example of some of the following Theorem 1 or Theorem 2 from $$f(n)=\frac{\sigma(n)}{n^2}$$ in Tao, ...
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2answers
61 views

If $b^2 \equiv 1 \pmod 3$, is it possible to have $\sigma(b^2) \equiv b^2 \pmod 3$?

The title says it all. Let $\sigma(N)$ denote the sum of the divisors of the positive integer $N$. To paraphrase my question: If $3 \mid \left(b^2 - 1\right)$, is it possible to have $3 \mid \...
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A question on the greatest common divisor of integers and their divisor sum

Suppose that $x, y, z$ are positive integers. Let $\sigma(x)$ be the sum of the divisors of $x$, and let $\gcd(y, z)$ be the greatest common divisor of $y$ and $z$. Here is my question: If the ...
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1answer
19 views

Can $\sigma(2^r)$ be abundant for $r > 1$?

Let $\sigma(x)$ be the sum of the divisors of the positive integer $x$. If $\sigma(y) < 2y$, $y$ is called deficient; if $\sigma(z) > 2z$, $z$ is called abundant. Questions (1) Can $\...
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Doubts and computations about Dirichlet series and aliquot sequences I

Perhaps the more easier statement that one can deduce for aliquot sequences (which is the Wikipedia's Page) is the following Lemma. For an integer $n\geq 1$, let $s^0(n)\equiv n$, $s(n)\equiv s^1(...
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2answers
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Find all the natural numbers $n$ such that $\sigma(n)=15$

Find all the natural numbers $n$ such that $\sigma(n)=15$ Where $\sigma (n)$ is the sum-of-divisors function My attempt: $$n=p_1^{\alpha_1}\cdots p_s^{\alpha_s}$$ $$\sigma(n)=\frac{p_1^{\...
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3answers
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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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65 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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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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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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22 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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152 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. $$ P(N,x)=\sum_{n=2}^{N}\frac{1}{n}\...
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1answer
62 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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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 properties ...
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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 Möbius 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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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
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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) $$1^2+2^2+\ldots+n^2=\frac{n(n+1)(...
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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
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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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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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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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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
34 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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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
34 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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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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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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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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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
31 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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52 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 n(2-m)k}{m}\...
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1answer
16 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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23 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 ...