Questions tagged [divisor-sum]

For questions on the divisor sum function and its generalizations.

In mathematics, and specifically in number theory, a divisor function is an arithmetic function that returns the number of distinct positive integer divisors of a positive integer.

Definition: The divisor function is the sum of positive integers dividing $n$, i.e., $$\sigma(n)=\sum\limits_{d\mid n} d~.$$ As usual, the notation "$d \mid n$ " as the range for a sum or product means that $~d~$ ranges over the positive divisors of $~n~$.

Often a related, more general function $$\sigma_a(n)=\sum\limits_{d\mid n} d^a$$ is studied. Both these functions are multiplicative.

The number of divisors function is given by $$\tau(n)=\sum\limits_{d\mid n} ~1$$

For example, the positive divisors of $~15~$ are $~1,~ 3,~ 5,~$ and $~15~$. So $$\sigma (15)=1+3+5+15=24\qquad \text{and}\qquad \tau(15)=4~.$$

References:

https://en.wikipedia.org/wiki/Divisor_function http://mathworld.wolfram.com/DivisorFunction.html http://sites.millersville.edu/bikenaga/number-theory/divisor-functions/divisor-functions.html

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Does there exist $a_0$, such that $\{a_n\}_{n=0}^{\infty}$ is unbounded?

Suppose $\{a_n\}_{n=0}^{\infty}$ is a sequence, defined by the recurrence relation $$ a_{n+1} = \phi(a_n) + \sigma(a_n) - a_n, $$ where $\sigma$ denotes the divisor sum function and $\phi$ is Euler's totient function. Does there exist $a_0$ such…
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Are there any natural numbers $n$ that satisfy the condition $7921\sigma(n) = 15840n$?

Are there any natural numbers $n$ that satisfy the condition $7921\sigma(n) = 15840n$, where $\sigma(n)$ denotes the sum of divisors of $n$? This question arises from the theory of immaculate groups (or, equivalently, Leinster groups). An immaculate…
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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). Clearly $A_n \subseteq \{ 1,2,3,\ldots,n\}$ (since…
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What is the sum of all positive even divisors of 1000?

I know similar questions and answers have been posted here, but I don't understand the answers. Can anyone show me how to solve this problem in a simple way? This is a math problem for 8th grade students.Thank you very much! What is the sum of all…
learning
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When $p$ is an odd prime, is $(p+2)/p$ an outlaw or an index?

Let $\sigma(x)$ denote the sum of the divisors of $x$, and denote the abundancy index of $x$ as $$I(x) = \dfrac{\sigma(x)}{x},$$ and the deficiency of $x$ as $$D(x) = 2x - \sigma(x).$$ If the equation $I(a)=b/c$ has no solution $a \in \mathbb{N}$,…
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The equation $\sigma(n)=\sigma(n+1)$

In OEIS, the solutions of $$\sigma(n)=\sigma(n+1)$$ where $\sigma(n)$ denotes the sum of the divisors of $n$ including $1$ and $n$ , are shown upto $n=10^{13}$ The entry can be found already by entering the first three solutions $14,206,957$ It is…
Peter
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Proving that $\sigma_7(n) = \sigma_3(n) + 120 \sum_{m=1}^{n-1} \sigma_3(m)\sigma_3(n-m)$ without using modular forms?

This problem appears as a (starred!) exercise in D. Zagier's notes on modular forms. I have to admit that I have no idea how to do it. Here, $\sigma_k(n) =\sum_{d\mid n} d^k$, as usual. This identity is traditionally obtained by using the fact that…
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$\sigma(n) \equiv 1 \space \pmod{n}$ if and only if $n$ is prime

For $n>1$, let $\sigma(n)$ denote the sum of all positive integers that evenly divide $n$. Then $\sigma(n) ≡ 1 \space(mod\space n)$ if and only if $n$ is prime. I've been trying to prove this for a long time, but i can't figure it out. I found a…
user532929
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An integer is prime iff $\phi(n) \mid n-1$ and $n+1 \mid \sigma (n)$

I wish to prove An integer is prime iff $\phi(n) | n-1$ and $n+1|\sigma (n)$ where $\phi$ is Euler's totient function and $\sigma(n)$ is the sum of the positive divisors of n. I can show from a previous exercise that $\phi(n)|n-1$ implies n is…
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Are there infinitely many sets of relatively prime numbers with equal number and sum of divisors?

Consider the prime factorization of the numbers $14$ and $15$ : $$14 = 2 \cdot 7 \implies \tau(14) = 2 \cdot 2 = 4 \space ;\space \sigma(14) = 3 \cdot 8 = 24$$ $$15=3 \cdot 5 \implies \tau(15) = 2 \cdot 2 = 4 \space ;\space \sigma(15) = 4 \cdot 6 =…
Haran
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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 $\frac{1}{d_1}+\frac{1}{d_2}+\cdots + \frac{1}{d_k}$. Attempt: Consider, $d_1, d_2,…
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Proving $ \frac{\sigma(n)}{n} < \frac{n}{\varphi(n)} < \frac{\pi^{2}}{6} \frac{\sigma(n)}{n}$

This is an exercise from Apostol's number theory book. How does, one prove that $$ \frac{\sigma(n)}{n} < \frac{n}{\varphi(n)} < \frac{\pi^{2}}{6} \frac{\sigma(n)}{n} \quad \text{if} \ n \geq 2$$ I thought of using the formula $$\frac{\varphi(n)}{n}…
anonymous
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Does the equation $\sigma(\sigma(x^2))=2x\sigma(x)$ have any odd solutions?

(Note: This question has been cross-posted to MO.) Let $\sigma(x)$ denote the classical sum of divisors of the positive integer $x$. Here is my question: Does the equation $\sigma(\sigma(x^2))=2x\sigma(x)$ have any odd solutions? MY ATTEMPT I…
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Is there an odd solution of $\varphi(n)+n=\sigma(n)$?

I want to show that the only solution of $$\varphi(n)+n=\sigma(n)$$ for a positive integer $n$ is $n=2$. What I worked out is that we must have $$\varphi(n)>\frac{n}{2}$$ To show this assume $n$ is composite and its prime factors are…
Peter
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Interesting convergence in divisor sums up to $10^k$

Let $S(k)$ be the sum of divisors across each of $1, 2, ..., 10^k$. For example, $$\begin{align}S(1) &= 1 + 3 + 4 + 7 + 6 + 12 + 8 + 15 + 3 + 17 \\&= 87\end{align}$$ where each of $1 + 3 + ... + 17$ are the sums of divisors of $1, 2, ..., 10^1$…
Yiyuan Lee
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