For questions on the divisor sum function and its generalizations.

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2
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1answer
23 views

How to calculate $\sigma(x^k)$?

If $n = 2^k$, then $\sigma(n) = 2 \times 2^k - 1$, For example: $$\sigma(28) = \sigma(2^2) \times \sigma(7) = 7 \times 8 = 56$$ But how about: $$\sigma(100) = \sigma(2^2) \times \sigma(5^2) = 7 ...
1
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1answer
18 views

Question on Sum of Divisor?

I know $\sigma(m)=24$ for $m=\{14,15,23\}$ but how can we find this numbers? Here is what I did Let the prime factorization of $m$ be $$m=p_1 ^{\alpha _{1}}p_2 ^{\alpha _{2}}\cdot\cdot\cdot p_k ...
1
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1answer
31 views

Determine if $\sum_{q=1}^{\lceil n/2\rceil}R_q(n)$ gives the number of divisors of $n$.

Let $$R_q(n)=\left\{\begin{array}{lll} r\left(\dfrac n{2q-1}\right)&\text{if }(2q-1)\mid n\\ 0&\text{otherwise}\end{array}\right\},$$ where $r(n)$ is the ruler function, i.e., the $2$-adic ...
1
vote
0answers
29 views

Prove a property of the divisor function (Part 2)

Further to this MSE question, I would like to pose a follow-up inquiry: If $n \in \mathbb{N}$ and $(\sigma(n) - n) \mid (n - 1)$, does it follow that $n$ and $\sigma(n)$ would have to be coprime, so ...
3
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0answers
96 views

Divisor summatory function for squares plus one

As an exercise for my Analytic Number Theory course, I need to prove, using Dirichlet hyperbola method, that: $\sum_{n\leq x}\tau(n² + 1)= {3\over\pi}x\log(x) + O( x)$, where $\tau(n)=\sum_{d|n}1$ ...
2
votes
1answer
179 views

Proof regarding Robin's inequality (RI).

Let $\sigma$ be the divisor sum function, $\gamma$ the Euler-Mascheroni constant and $n>5040$. Robin showed that if the inequality$$\displaystyle \sigma(n)<e^{\gamma}n\log\log n$$ ever fails, it ...
0
votes
1answer
29 views

Function $f(n) = 2^{\omega(n)}\mu^2(n)$

Let $$f(n) = 2^{\omega(n)}\mu^2(n)$$, where $\omega(n)$ is number of distinct prime divisors of $n$ and $\mu(n)$ is Moebius function. I want to simplify it. As long as $$ ...
0
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1answer
34 views

About the sigma function and an interesting inequality. [closed]

Is it true $\sigma(A)$/$\sigma(B)$ > = (A/B) ; given B divides A ?
2
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0answers
63 views

Whats wrong in this proof of $10$ is a solitary number?

Friendly numbers are two or more natural numbers with a common abundancy, the ratio between the sum of divisors of a number and the number itself. Two numbers with the same abundancy form a friendly ...
1
vote
1answer
21 views

how to sum the divisors of N mod K if all I have is N mod K?

The input to this problem is N. I have to calculate 2 things: 1 - N! mod (10^9 + 7) 2 - sum of all divisors of N! mod (10^9 + 7) I know how to do the first step, I'm wondering if there is a way ...
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2answers
70 views

About the 'sigma' function.

Is it true that if $n$ divides $m$ , $\sigma(\frac mn) \leq \frac{\sigma(m)}n$. If so this has a bearing on counterexamples to Robin's inequality.
0
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0answers
22 views

Sum of convolution of divisor function [duplicate]

For every integer $k$ let $d_k: \mathbb{N} \rightarrow \mathbb{C}$ be defined recursively as $d_0 = \mathbf{1}$, $d_k = d_{k-1} * \mathbf{1}$. So for example $d_1 (n) = d (n) = \sum_{d \vert n} 1$ is ...
0
votes
1answer
40 views

Why is this Moebius equivalence true?

I would like to know why the following is true: $$\tau(n^2) = \sum_{d | n} \mu(n/d)(\tau(d))^2$$ I cannot derive it. It is on OEIS but I'd like to know how this was found. $\tau(n)$ is the count of ...
1
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0answers
41 views

Number of divisiors of $n$ less than $m$

I'm looking for a closed- or alternative-form of the function that counts the number of divisors of an integer $n$ that are less than some integer $m$ (interested in $m < n$, obviously): $ ...
0
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1answer
86 views

Who should I ask for Robin's paper? At any rate, I want to find out if a similar result to his can be achieved with 36 instead of 12.

Robin proved unconditionally that for $\ n \ge 3$ , $$ \sigma(n)<\left(e^\gamma+{\log\log12\left({\frac73}-e^\gamma \log\log12\right)\over (\log \log n)^2}\right)n \log \log n. $$ I need a similar ...
0
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0answers
46 views

If Lagarias' inequality is wrong, are there infinitely many counterexamples to it?

I do know that since Robin's (RI) and Lagarias' (LI) inequalities are both equivalent to RH, they're also equivalent one another, hence if RI is false, so is LI. And Robin proved there are infinitely ...
0
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0answers
22 views

Could you give a definition of what is a superior highly composite number using only words?

I know very well what is a superior highly composite number, but I would like to see how could we (roughly) define what is a superior highly composite number using only words (using no equations and ...
0
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0answers
57 views

With odd $\ n>9, \sigma(n) < {11\over 16} e^{\gamma} n \log \log n $?

If that's not the case, do we know anyway some upper bound better than that given by Robin's inequality, since it has been shown that it holds for all odd numbers > 9 ( Choie, YoungJu, et al. "On ...
1
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0answers
73 views

Find a closed form for the constant term

In a previous question, an asymptotic expansion was provided for the weighted divisor summatory function $\displaystyle \frac {d(n)}{n}$: $$\sum_{n\leq ...
0
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1answer
47 views

Why are multiples of abundant numbers also abundant numbers?

I am currently working on Project Euler Problem 23 which involves abundant numbers. In short, abundant numbers are numbers that are less than the sum of their proper divisors. For example, 12 has ...
0
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0answers
42 views

What are the patterns in the number of divisors $d(n)$ of the highly composite numbers?

I am trying to understand the patterns in the number of divisors $d(n)$ of the highly composite numbers. The numbers marked with an asterisk are the superior highly composite numbers. The first ...
0
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0answers
52 views

Given an integer n find smallest integer i such that σ(i)=n. Smallest Inverse Sum of Divisors

Hi All I need some help I am trying to solve this problem which involves computation of sum of divisors and its inverse. In other words Given an integer n find smallest integer i such that σ(i)=n ...
3
votes
2answers
64 views

Infinite series involving sum of divisors function

Is there much known about infinite series of the form $\sum_{n=1}^{\infty}\frac{\sigma_{1}(n)}{n}q^{n}$ where $\sigma_{1}(n)$ is the sum of divisors function. I am particularly interested in ...
1
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1answer
78 views

the average order of divisor function

In Analytic number theory by Apostol there's a theorem: $$\sum_{n\le x} \sigma(n)= \frac{1}{2} \zeta(2)x^2 + O(x\log x)$$ and then it claims that because we know that $\zeta (2)= \frac{\pi^2}{6} $ ...
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1answer
35 views

Prove that for any $n$ having at least two distinct prime factors, there exists a perfect square between $n$ and $\sigma(n)$

I came up with a new problem, described as follows: Prove that for any $n$ having at least two distinct prime factors, there exists at least one perfect square between $n$ and $\sigma(n)$ I will post ...
1
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0answers
33 views

Can we prove this inequality in another way?

As explained here, I've managed to prove the following inequality: $\sigma(n)\geq\sqrt n(d(n)-2)+n+1$. This can be proved easily in two cases (one for $n$ being a perfect square and one for otherwise) ...
4
votes
1answer
261 views

Is there a closed form expression for the sum of all the proper divisors of an integer?

I have already found a summation formula here: http://math.stackexchange.com/a/22723, and also a very interesting recursive formula here: http://math.stackexchange.com/a/22744. Any ideas on how to ...
0
votes
1answer
41 views

What are the smallest numbers $n$ such that $\dfrac{d(n)}{\ln(n)} \geq k$ where $d(n) = \sigma_0(n)$ is the number-of-divisors function?

I have calculated $\dfrac{d(n)}{\ln(n)}$ on a few highly composite numbers up to 5040. Here is what I got: $\dfrac{d(120)}{\ln(120)} = 3.3420423$ $\dfrac{d(360)}{\ln(360)} = 4.0773999$ ...
1
vote
1answer
44 views

Prove that $\dfrac{\sigma_1(n)}{n} = \sigma_{-1}(n)$ where $\sigma_x(n)$ is the sum of the $x$th powers of the positive divisors of $n$.

I computed $\dfrac{\sigma_1(n)}{n}$ and $\sigma_{-1}(n)$ on a good hundred values of $n$, and they seem to always match. For example: $\dfrac{\sigma_1(6)}{6} = \dfrac{1 + 2 + 3 + 6}{6} = ...
0
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1answer
43 views

A generalization of perfect numbers

In analogy to perfect numbers, let's say I wanted a set $S = \{s_1, s_2,..., s_n\}$ of numbers with the property: $$\frac{s_1+s_2+\cdots s_n}{n}= \sigma(S) := \sum_{d\mid s_1\wedge d\mid s_2, d\mid ...
2
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1answer
84 views

Holomorphic Differentials on a non-singular curve.

So I've been working on this for an exam I have coming up and I'm not sure I really understand. If I have a curve defined by some homogenous polynomial P, I can show that the canonical divisor class ...
0
votes
1answer
121 views

Why does the number of divisors of a superior highly composite number is always a highly composite number up to 720720 ? (the only exception is 120)

I've calculated the number of divisors of every superior highly composite number up to $10^{27}$: http://oi59.tinypic.com/ndaijo.jpg The number of divisors of a superior highly composite number is ...
1
vote
1answer
192 views

Why isn't $1$ a superior highly composite number?

A superior highly composite number is a positive integer $n$ for which there is an $\epsilon>0$ such that $\dfrac{d(n)}{n^\epsilon} \geq \dfrac{d(k)}{k^\epsilon}$ for all $k>1$, where the ...
1
vote
2answers
29 views

How to show $\sum_{n\leq x}d(n)=\sum_{ab\leq x}1$?

How to show this equation below is true. $$\sum_{n\leq x}d(n)=\sum_{ab\leq x}1$$ $d(n)$ is the divisior function. It seems easy but i just can't see it.
1
vote
1answer
51 views

How to prove $D(n)<2n(\log\log n)$?

How to prove $D(n)<2n(\log\log{n})$ for all sufficiently large $n$ where $D(n)$ is the Divisor summatory function.
0
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0answers
98 views

Among the superior highly composite numbers, which are the most divisor dense numbers?

I’m searching for the most divisor dense natural numbers. Firstly we have the highly composite numbers: 1, 2, 4, 6, 12, 24, 36, 48, 60, 120, 180, 240, 360, 720, 840, 1260, 1680, 2520, 5040, … But ...
3
votes
0answers
41 views

When is $f(n)=\sum\limits_{d\mid n}\sigma(d)$ prime?

When is $f(n)=\sum\limits_{d\mid n}\sigma(d)$ prime? Note, $f$ is multiplicative and $\sigma(n)>1, \;n>1$. Therefore $f(n)$ is prime only when $n=p^\alpha$, with $p$ prime, $\alpha\geq1$. ...
4
votes
1answer
94 views

How can I prove that $\frac{\sigma(n)}{n} = \sum_{(d|n)} \frac{1}{d}$ for every $n \in \mathbb{Z^{+}}$?

I want to show that $\displaystyle \frac{\sigma(n)}{n} = \sum_{(d|n)} \frac{1}{d}$ for every $n \in \mathbb{Z^{+}}$. This is essentially a basic number theory question. I am able to get to the ...
3
votes
1answer
58 views

Explanation for a theorem pertaining on Dirichlet character sums

A very well known theorem pertaining on Dirichlet characters sums states that if $\chi$ is a Dirichlet character modulo $k$, defining $$ A\left(n\right)=\sum_{d\mid n}\chi\left(d\right) $$ Then ...
2
votes
3answers
203 views

How to find the sum of $k$th powers of all proper divisors of first $n$ numbers

I am trying this problem but unable to come up with efficient algorithm can someone help with this problem. I have solved the easier version of the problem below is the problem link. Thanks in ...
0
votes
1answer
20 views

Is $N$ considered a proper divisor of $N$

I just started Project Euler and I have run into an issue. There are numerous problems that ask about the proper divisors, sum of proper divisors, etc. of a number $N$. Now, before I proceed, I would ...
7
votes
0answers
111 views

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 ...
0
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0answers
62 views

Question on a “proof”

Let $\sigma(x)$ be the sum of divisors of the positive integer $x$. Define the abundancy index $I$ as $I(x) = \sigma(x)/x$. Suppose that we have the condition $$I(a^2)I(b^c) = 2.$$ Let $a$ and $c$ ...
0
votes
1answer
47 views

How Can I find the summation of divisors of $n^p$.

For Example $n=8$ and $p=2$. So $n^p=64$. And the summation of divisors is $1+2+4+8+16+32+64=127$. But the problem arises when $n=10^6$ and $p=10^6$. Remember u can modulus the result by $100$.
6
votes
0answers
78 views

Does there exist a positive $k$ s.t. for all $r\geq k$, “$\sigma_r(m)<\sigma_r(n)$ for every $m<n$” for infinitely many odd positive integers $n$?

Does there exist a positive real $k$ such that for all real $r\geq k$, "$\sigma_r(m)<\sigma_r(n)$ for every $m<n$" for infinitely many odd positive integers $n$? $\sigma_r(n)$ is the sum of the ...
1
vote
1answer
116 views

pairwise disjoint subsets of divisors of $ n $ (maximum number)

Let $ n \in \mathbb{N} $, $ n>1 $ and $ a_1,\ldots,a_k \in \mathbb{N} $ (not necessarily distinct!) with $ a_i \mid n $ for all $ i=1,\ldots,k $ be given. Assume that $ \sum_{i=1}^k a_i = K\cdot n ...
0
votes
2answers
42 views

Is the following true or not?

I have found something like this: $(((a^{x}-1)mod\ p)* ( a-1) ^ {p-2})mod\ p = \frac{a^{x}-1}{a-1} mod \ p $ After taking some examples and considering the place I took this from this should be ...
1
vote
2answers
65 views

Explain the origin of the number of divisors and sum of divisors formulas.

I know the basic formulas which are: For a number $n = p_1^{a_1} p_2^{a_2} \cdot \ldots \cdot p_k^{a_k}$, we have $d(n) = ( a_1 + 1 )( a_2 + 1 ) \cdot \ldots \cdot (a_k+1)$ and $S(n) = ...
1
vote
1answer
88 views

Bounds on the average of the divisors of natural numbers.

I managed to find an interesting inequality containing the sum of the divisors of a number and the number of them, using AM-GM between each 2 of the divisors of it: $\sigma(n)\geq\sqrt n.d(n)+(\sqrt ...
1
vote
0answers
189 views

Ramanujan Sum and the sum of divisors

How would one prove that $\sigma(n) = \frac{\pi^2}{6}n\sum_{q = 1}^\infty q^{-2}c_q(n)$ where $c_q(n) = \sum_{a = 1, (a, q) = 1} e(an/q)$ and $\sigma(n)$ is the sum of the divisors of $n$? Also, how ...