The tag has no usage guidance.

learn more… | top users | synonyms

17
votes
4answers
2k views

Have I found all the numbers less than 50,000 with exactly 11 divisors?

The math problem I am trying to solve is to find all positive integers that meet these two conditions: have exactly 11 divisors are less than 50,000 My starting point is a number with exactly 11 ...
1
vote
0answers
18 views

Can I presume that this inequality is a good aproximation for a divisor function?

I've used the Lemma 7.9 from page 73 from Krizek, Luca and Somer, 17 Lectures on Fermat Numbers From Number Theory to Geometry Springer CMS (2001) (you can see this page as a Google Book, type here ...
6
votes
2answers
137 views

Number of solutions to this nice equation $\varphi(n)+\tau(n^2)=n$

How many natural numbers $n$ satisfy the equation$$\varphi(n)+\tau(n^2)=n$$where $\varphi$ is the Euler's totient function and $\tau$ is the divisor function i.e. number of divisors of an integer. I ...
0
votes
2answers
27 views

On those integers $n>1$ such that there exists a commutative ring with identity with exactly $n$ ideals

Let $n>1$ be an integer; we call $n$ a "ring number" if there exists a commutative ring $R$, with identity, having exactly $n$ ideals (including $\{0\}$ and $R$); now since for every $n>1$, $\...
4
votes
3answers
54 views

Sum of number-of-divisors function equals $\sum_{j=1}^{n} \lfloor n/j \rfloor$.

I am trying to prove the identity $$t(1) + t(2) + \cdots + t(n) = \Big\lfloor \dfrac{n}{1} \Big\rfloor + \Big\lfloor \dfrac{n}{2} \Big\rfloor + \cdots + \Big\lfloor \dfrac{n}{n} \Big\rfloor,$$ where ...
1
vote
1answer
50 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, $...
1
vote
1answer
118 views

Possible divisors of $s(2s+1)$ follow up question.

This question is related to this post:Possible divisors of $s(2s+1)$. I have some follow up questions which should be a new post. I write $\psi(s) = s(2s+1)$. We showed that for every prime $s$ that ...
2
votes
2answers
56 views

Possible divisors of $s(2s+1)$

I write $\psi(s) = s(2s+1)$ and let $d$ be the divisor function. If $s$ is prime then 4 divides $d(\psi(s))$. For example if $s=37$ then $d(\psi(s)) = d(2775) = 12$ and $4|12$. Is this trivial? I am ...
1
vote
1answer
14 views

find number of positive integers smaller $N$ that have an even number of proper divisors

Given a number $N$, I want to determine $\vert\{x\in\mathbb{N}\ \vert\ 1\leq x\leq N\wedge\ \text{candidate}(x)\}\vert$ where $\text{candidate}(x) = \vert\{y\in\mathbb{N}\ \vert\ 1\leq y < x \...
3
votes
0answers
19 views

A function related to divisior counting function

Let $d(n)$ be the divisor function. Let $d_{2}(n)=d(d(n))$, $d_{3}(n)=d(d(d(n)))$, $d_{4}(n)=d(d(d(d(n))))$ and so on... We're gonna define $f(n)$, the smallest number satisfies $d_{f(n)}(n)=2$. For ...
1
vote
1answer
34 views

$\sigma _{0}(n)=\sigma _{0}(n+1)$ will occur infinitely often. [closed]

In 1984, Roger Heath-Brown proved that will occur $\sigma _{0}(n)=\sigma _{0}(n+1)$ infinitely often. How did he prove that? I couldn't find the paper on the internet.
0
votes
0answers
24 views

Questions about $w(\prod_{k=1}^{n}(p_{k}-1))-w(\prod_{k=1}^{n}(p_{k}+1))$

Let's define this functions. $$f(n)=\prod_{k=1}^{n}(p_{k}-1)$$ $$g(n)=\prod_{k=1}^{n}(p_{k}+1)$$ $$h(n)=\mid w(f(n))-w(g(n))\mid $$ where $p_{k}$ is $k$th prime number and $w(n)$ gives the number of ...
1
vote
2answers
33 views

When the Sum of digits exceed the Number of Divisors

Could somebody help me prove that there are a infinite number of natural numbers for which their sum of digits exceeds the number of divisors? If $S(n)$ denoted the sum of digits, and $\sigma_k(n)$ ...
0
votes
1answer
34 views

Working with divisors [closed]

Compute βˆ… (40), 𝜎(124), 𝑑(124) and check the equality in Ξ£βˆ…(𝑑) = 40. Here's what I've done so far: Not really sure about the summation equality. βˆ… (40) = βˆ… (5)...
0
votes
1answer
140 views

Average number of square free divisors for $n\leq x$

Let $d_{sf}(n)$ be the number of square-free divisors of $n$, and let $D_{sf}(k)=\sum_{n=1}^{k} d_{sf}(n)$ denote the corresponding summation function. Mertens showed that the asymptotic expansion of ...
-1
votes
1answer
36 views

Counting divisors of a number

Let m be any positive integer and consider $\Sigma_{d|m} \frac{1}{d} $. I wish to ask whether there is a closed form expression for the above sum.
0
votes
0answers
15 views

What is the intuitive idea behind this claim on the divisor function?

I want to understand the claim $$ \lim_{X \to \infty} \frac{\#\{n \le X : d(n) \equiv 0 \mod m \}}{X} = 1, $$ where $m \in \mathbb{Z}^+$ is fixed and $d(n)$ is the divisor function. I think that it ...
0
votes
0answers
25 views

What number from an integer range has the most divisors? [duplicate]

I've been wondering. What number from an integer range (-2 147 483 647/+2 147 483 647) has the most divisors and how many is that?
0
votes
1answer
65 views

Is the asymptotic growth rate of the product of divisor function up to $n$ known?

Let $\tau(k)$ be the number of divisors of the positive integer $k.$ How does $f(n)\stackrel{\triangle}{=}\prod_{k\leq n} \tau(k)$ or a reasonable function of it,such as $\log f(n)$ or $f(n)^{1/n}$ ...
6
votes
0answers
135 views

From $\prod_{d\mid n}d=n^{\sigma_0(n)/2}$ to $n!=\operatorname{lcm}(1,\ldots,n)^{e(n)}$, where $\sigma_0(n)$ is the number of divisors

We know that $$\prod_{d\mid n}d=n^{\sigma_{0}(n)/2}$$ for every integer $n\geq 1$, where $\sigma_{0}(n)$ is the number of positive divisors of $n$, see for example [1] (exercise 10, page 47). And for ...
0
votes
1answer
77 views

Injective function? Find a formula? Factors [closed]

I need some help with the following question, thanks. Let $$r : \mathbb N β†’ \mathbb N$$ be the function where the output is the number of positive integer factors of the input. For example, $$r (15) ...
5
votes
0answers
252 views

A natural number has exactly 10 divisors including 1 and itself.How many distinct prime factors can this natural number have?

A natural number has exactly 10 divisors including 1 and itself.How many distinct prime factors can this natural number have? options given: (a) either $1$ or $2$ (b)$1$ or $3$ (c)either $2$ or $3$ ...
6
votes
2answers
181 views

A $q$-continued fraction connected to the divisor function?

In this post, the following two continued fractions discussed by Nicco are given, $$A(q)= \left(\frac{\vartheta_2(0,q)}{2\,q^{1/4}}\right)^2= \cfrac{1}{1-q+\cfrac{q(1\color{red}-q)^2}{1-q^3+\cfrac{q^...
2
votes
1answer
44 views

Prove or disprove: $ \sum_{b \vee d = x} \tau(b) \tau(d) = \tau(x)^3$

Can somebody prove or disprove? Let $\tau$ be the divisors function, so that $\tau(6) = \#\{ 1,2,3,6\} = 4$ $$ \sum_{b \vee d = x} \tau(b) \tau(d) = \tau(x)^3$$ Here I am using $b \vee d = \mathrm{...
3
votes
1answer
58 views

The largest subset of a finite cartesian product in which distinct elements differ in at least 2 components

Let $A_1,\ldots,A_n$ be finite sets of sizes $a_1,\ldots,a_n$. What is the largest possible size of a subset $S\subset\bigotimes A_k$ such that if $(d_1,\ldots,d_n),(e_1,\ldots,e_n)\in S$, then $\...
0
votes
2answers
147 views
0
votes
2answers
116 views

Probability a string has $2$ digits, $4$ consonants, and $1$ vowel, given a length of $7$ w/o repetition

My thinking behind this problem would be to pick $2$ digits out of 10 total $\binom{10}{2}$, $4$ consonants out of $21$, vowels not included $\binom{21}{4}$, and the $5$ vowels, multiplied by $7!$. $...
0
votes
1answer
28 views

Number Theoretic Sum of three variables. Having trouble isolating two of them.

I encountered the following sum: $$\frac{c_{mn-p}}{c_0} = \sum_{d|p}\ \frac{a_{n-d}b_{m-\frac{p}{d}}}{a_0b_0} $$ Where $$a_i=0 \text{ when } i<0 \text{ and } b_j=0\text{ when } j<0\text{ and ...
8
votes
2answers
135 views

Show $\forall\varepsilon>0\,\lim_{n\to\infty}\frac{\#\{\text{positive divisors of n}\}}{n^\varepsilon}=0$

Show that $\forall\varepsilon>0,$ $$\lim_{n\to\infty}\frac{\#\{\text{positive divisors of n}\}}{n^\varepsilon}=0$$ I'm trying to solve this problem for a long time, but I'm really stuck I have ...
16
votes
1answer
431 views

Numbers having in decimal representation no common digits with all their proper divisors

Let us call a positive integer having in decimal representation no common digits with all its proper divisors "a good number". $54$ is a good number : $1,2,3,6,18,27$ $48$ is not a good number : $1,...
5
votes
1answer
217 views

Euler's totient and divisors count function relationship when $[(\frac{\varphi(n)}{2}+1)\cdot(\frac{\tau(n)}{2}+1)] = n$

I am studying the Euler's totient function $\varphi(n)$ and the divisors count function, $\tau(n)$, also named $d(n)$, and recently opened a question (link here) about the following condition: (1)$...
4
votes
1answer
50 views

Number of Divisors of most numbers

In the book A Comprehensive Course in Number Theory by Alan Baker. The author mentions that even though the average order of $\tau(n)$ is $\log n$, almost all numbers have about $(\log n)^{\log 2}$ ...
0
votes
1answer
34 views

Analogue to superior highly composite numbers for the unitary divisor function

For which positive integers $n$ does there exist a positive real number $\epsilon$ such that $\dfrac{2^{\omega(n)}}{n^\epsilon}\geq\dfrac{2^{\omega(k)}}{k^\epsilon}$ for all $k<n$, and $\dfrac{2^{...
2
votes
1answer
32 views

proofs of $n\sum_{d|n}\frac{(-1)^{d+1}}{d}=\sum_{d|n,d \text{ odd}}d$

I'm looking for other proofs of the identity : For all integer $n\in \mathbb{N}^{+}$: $$n\sum_{d|n}\frac{(-1)^{d+1}}{d}=\sum_{d|n,d \text{ odd}}d $$ where in the first sums is taken over all ...
-1
votes
1answer
59 views

Find an $n$ not a power of a prime such that $n$ has 51 positive divisors [closed]

Find an $n$ not a power of a prime such that $n$ has 51 positive divisors. I'm not sure where to even start with this question. Any help would be really appreciated.
0
votes
1answer
82 views

$d(n)$ is odd iff $n = k^2$ [duplicate]

The function $d(n)$ gives the number of positive divisors of $n$, including $n$ itself. For example, $d(25) = 3$ because $25$ has three divisors: $1$, $5$, and $25$. Prove that $d(n)$ is odd if and ...
2
votes
2answers
356 views

Product of Divisors of some $n$ proof

The function $d(n)$ gives the number of positive divisors of $n$, including n itself. So for example, $d(25) = 3$, because $25$ has three divisors: $1$, $5$, and $25$. So how do I prove that the ...
2
votes
1answer
284 views

Can someone help me prove that $\tau(n)$ is odd iff $n$ is a perfect square. [duplicate]

Can someone help me prove that $\tau(n)$ is odd if and only if $n$ is a perfect square. So basically I have to prove that $\tau(n)$ is odd iff $n = k^2$ for some integer $k$. $\tau(n)$ is the ...
6
votes
1answer
318 views

The number of divisors of a number whose sum of divisors is a perfect square

Let $n$ denote a non-prime whose sum of divisors is a perfect square. I have noticed a few surprising facts on the number of divisors of $n$: It is either prime or semi-prime or $27$ in all cases ...
3
votes
4answers
871 views

How many number between 1 and 1000 satisfy a certain condition?

How many positive integers less than $1,000$ are multiples of $5$ and are equal to $3$ times an even integer? It is simply asking for multiples of $5$ and $6$ Is there a way to do this without ...
2
votes
1answer
92 views

express the dirichlet series for the sequence d(n)^2 in terms of riemann zeta.

Prove that $$\sum_{n=1}^\infty d(n)^2n^{-s}=\zeta(s)^4/\zeta(2s)$$ for $\sigma>1$ what i did: I already proved this formally, that is, without considering convergence. I use euler products, ...
4
votes
1answer
51 views

Good (asymptotic) upper bound for recurrence over divisors

I am looking for a good (asymptotic) upper bound on the following recurrence relation ($T(0) = 1)$: $$ T(n) = \left[ \sum_{1\leq d<n, d|n} T(d) \right] + 1 $$ Note that the recursion is only for ...
0
votes
1answer
37 views

Prove the divisors pairs

If we arrange the positive distinct divisors of a number A by increasing order, then we get something like: $$1<a_1<a_2<a_3<...<a_{n-2}<a_{n-1}<a_n<A$$How can we prove that $$...
0
votes
2answers
75 views

How do I count all values that satisfy X mod N=1 in the range [A,B]

I want to count how many values of x in range [A,B] give remainder of 1 when divided by N. Is there any formula I can apply?
1
vote
2answers
90 views

Is there a way to show that $d(n)$, which counts the number of divisors of $n$ is non-increasing? [closed]

Is there a way to show that $d(n)$, which counts the number of divisors of $n$ is non-increasing? I'm trying to use the Cauchy condensation test to show that $\sum_{n\ge{2}}\frac{d(n)}{n\log^2n}$ is ...
2
votes
4answers
90 views

How to show that $\prod_{d/n} d = n^{\frac{\tau(n)}{2}}$ [duplicate]

set $ n, n \in \mathbb{N}$ and prove that $\prod_{d/n} d = n^{\frac{\tau(n)}{2}}$ ¨I have tried this¨ If $n > 1$ then $n = p_{1}^{\alpha_{1}}\cdot p_{2}^{\alpha_{2}}\cdots p_{k}^{\alpha_{k}}$ ...
5
votes
2answers
266 views

Prove that $\sum \limits_{d|n}(n/d)\sigma(d) = \sum \limits_{d|n}d\tau(d)$

How can I prove: $$\sum \limits_{d|n}(n/d)\sigma(d) = \sum \limits_{d|n}d\tau(d)?$$ Few observations : Left side is a sum function and the right side is a number of divisors function. Both the sides ...
7
votes
3answers
181 views

For any positive integer $n$, show that $\sum_{d|n}\sigma(d) = \sum_{d|n}(n/d)\tau(d)$

For any positive integer $n$, show that $\sum_{d|n}\sigma(d) = \sum_{d|n}(n/d)\tau(d)$ My try : Left hand side : $\begin{align} \sum_{d|p^k}\sigma (d) &= \sigma(p^0) + \sigma(p^1) + \sigma(p^2)...
6
votes
2answers
5k views

A number is a perfect square if and only if it has odd number of positive divisors

I believe I have the solution to this problem but post it anyway to get feedback and alternate solutions/angles for it. For all $n \in \mathrm {Z_+}$ prove $n$ is a perfect square if and only if $n$ ...
2
votes
1answer
75 views

The number of divisors of any positive number $n$ is $\le 2\sqrt{n}$

How to prove that the number of divisors of any positive number $n$ is $\le 2\sqrt{n}$? I have started something like below: $$ n^{\tau(n)/2} = \prod_{d|n} d$$ But not getting ideas on how to ...