For questions on the divisor sum function and its generalizations.

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1answer
27 views

Numbers and the largest sum of divisors [on hold]

Is there an algorithm to generate numbers that have the largest sum of divisors?
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47 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 ...
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1answer
22 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 ...
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0answers
28 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 ...
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0answers
24 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 ...
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2answers
56 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 ...
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1answer
61 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
30 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 ...
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0answers
30 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) ...
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1answer
225 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 ...
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1answer
39 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: $\dfrac{d(120)}{\ln(120)} = 3.3420423$ $\dfrac{d(360)}{\ln(360)} = 4.0773999$ ...
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1answer
35 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} = ...
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1answer
39 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 ...
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1answer
67 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 ...
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1answer
82 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 ...
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1answer
171 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 ...
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2answers
27 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.
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1answer
42 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.
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0answers
70 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 ...
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0answers
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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, $\alpha\geq1$. ...
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1answer
91 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
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1answer
53 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 ...
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3answers
111 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 ...
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1answer
18 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 ...
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104 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 ...
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60 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 $c$ ...
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1answer
43 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 $100$.
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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 ...
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1answer
114 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 ...
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2answers
40 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 ...
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2answers
59 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) = ...
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1answer
84 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 ...
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173 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 ...
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136 views

Sum of the unitary divisor function (mod p)

The sum of unitary divisors of a positive integer $n$ can be computed as $$ \sigma^\star(n) = \prod_{i=1}^{\omega(n)}(p_i^{a_{\large i}}+1) $$ where $\prod_{i=1}^{\omega(n)} p_i^{a_{\large i}}$ is the ...
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1answer
68 views

Is something similar to Robin's theorem known for possible exceptions to Lagarias' inequality?

Robin's theorem says that if $$\sigma(n)<e^\gamma n\log\log n$$ holds for all $n>5040$, where $\sigma(n)$ is the sum of divisors of $n$, then the Riemann hypothesis is true, but if there are any ...
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1answer
99 views

Does every record of the arithmetic derivative of natural numbers occur at a practical number?

Consider the arithmetic derivative of natural numbers, as defined here. By this definition, for every integer $n>1$, with canonical prime factorization ...
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0answers
39 views

please evaluate this sum

$$\sum^\infty_{n=1}\frac{(-1)^n}{n!}\times\frac{3^{a+n}}{a+n}-\sum^\infty_{n=1}\frac{(-1)^n}{n!}\times\frac{1}{a+n}$$ I need help evaluating the above, step by step would help. Thanks for a>0 or ...
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1answer
130 views

Simple Divisor Sum Transformation by Changing the Order of Double Summation

Show that $$\sum_{d|n} \frac{n}{d} \sigma(d) = \sum_{d|n} d \tau(d)$$ by changing the order of summations from each side to the other. $\sigma$ and $\tau$ are divisor sum functions. ...
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142 views

Simple Divisor Summation Inequality (with Moebius function)

Show that $$\left| \sum_{k=1}^{n} \frac {\mu(k)}{k} \right| \le 1 $$ where $\mu$ is Moebius function and n is a positive integer. The hard thing here is that the sum is not directly ...
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1answer
86 views

D.w. $p_i>\sigma(p_1^{a_1}p_2^{a_2}…p_{i-1}^{a_{i-1}})\forall i \in [1,\omega(n)]\iff d_j>d_1+d_2+…+d_{j-1} \forall j \in [1,\sigma_0(n)]$

Determine whether $(1):$ $$p_i>\sigma(p_1^{a_1}p_2^{a_2}...p_{i-1}^{a_{i-1}})\hspace{2mm}\forall i \in [1,\omega(n)]$$ is logically equivalent to $(2):$ ...
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51 views

How to divide total price into sum of multiple of numbers

How to divide total price into sum of multiple of numbers so that the sum is close to price as much as possible. Is there any formula? Example: Price: 1256.7 Numbers: 5.7, 3.6, 4.9, 10.1, 12.3, 6.8 ...
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0answers
113 views

On odd perfect numbers $N$ given in the Eulerian form $N = {q^k}{n^2}$

Note: This question was cross-posted from MO. Preamble: I apologize in advance if this particular MSE post would appear to be a bit of a polymath approach, I just had to put down all the details to ...
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1answer
77 views

Determine whether $\sigma(n)<e^\gamma n \omega(n)$ for all $n$ not of the form $2^x$

Determine whether $\sigma(n)<e^\gamma n \omega(n)$ for all $n$ not of the form $2^x$. In words (to define the symbols), the sum of the divisors of $n$ is less than the product of Euler's number to ...
3
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1answer
57 views

Is there a friendly pair $\{a, b\}$ where both $a$ and $b$ are odd?

Let $\sigma(x)$ denote the sum of the divisors of a positive integer $x$. $\{y, z\}$ is said to be a friendly pair if $$I(y) = I(z),$$ where $I(x) = \sigma(x)/x$ is the abundancy index of $x$. As ...
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0answers
27 views

Miranda's Exercise J Pag. 167

This is the exercise: If $v^{2}=h(u)$ defines a hyperelliptic curve of genus $g$, then $\phi=[1:u:u^{2}:\dots,u^{g-1}]$ defines a degree $2$ map onto a rational normal curve of degree $g-1$ in ...
2
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1answer
168 views

sum of divisors for given range of numbers from 1 to n

we are given a function F(n) for a number n which is defined as sum of the divisors of n (including 1 and n) ... now given an integer N we have to calculate G(n) = F(1) + F(2) + F(3) + ..... + F(n)... ...
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0answers
21 views

greater divior polynomials

Let $n,m$ be positive integers and $d = \gcd(n,m). $ Prove that $\gcd(x^{n} -1 ,x^{m} -1) = x^{d} -1$. Is this correct? Bezout : integers $r,s$: $\quad r(x^{n} -1) + s(x^{m} -1) = rx^{n} +sx^{m} - ...
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0answers
38 views

$\DeclareMathOperator*{\rad}{rad}$Prove that $\rad(\sigma(n)) \mid \rad(\sigma_0(n))$ iff $n$ is the product of one or more distinct Mersenne primes.

$\DeclareMathOperator*{\rad}{rad}$Prove that $\rad(\sigma(n)) \mid \rad(\sigma_0(n))$ if and only if $n$ is the product of one or more distinct Mersenne primes, where $\rad(k)$ is the product of the ...
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2answers
101 views

To prove $6|σ(6n-1) , ∀n∈ \mathbb N$

Let $σ(n)$ denote the sum of all the positive divisors of $n∈ \mathbb N$. I think that $6$ divides $σ(6n-1)$ for all $n∈ \mathbb N$ , but I am not able to prove it. So, a proof of the result (if it ...
0
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1answer
98 views

Does the sum of reciprocals of the harmonic divisor numbers converge?

Does the sum of reciprocals of the harmonic divisor numbers converge? Define the following: Harmonic divisor number - $n$ such that $\sigma(n) \mid n\sigma_0(n)$. Equivalently, the harmonic mean of ...