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0
votes
2answers
21 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
46 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
47 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
87 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
49 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
17 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
23 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 $\sigma_k(n)$ ...
1
vote
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… ...
0
votes
1answer
138 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
32 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
56 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
132 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 ...
1
vote
1answer
74 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
222 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 ...
5
votes
2answers
170 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= ...
2
votes
1answer
40 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 = ...
3
votes
1answer
57 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
120 views
0
votes
2answers
107 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 ...
0
votes
0answers
21 views

Function for the number of divisor of a number [duplicate]

Is there a formula/function that given any $n$ produces the number of divisors of $n$ ? And has that something to do with Euler function?
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
421 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 : ...
5
votes
1answer
202 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: ...
4
votes
1answer
49 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
32 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 ...
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 ...
0
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
75 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
214 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
268 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
265 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
837 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
86 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
49 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
36 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
74 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?
2
votes
2answers
89 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
85 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
243 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
170 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) + ...
6
votes
2answers
4k 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
70 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 ...
6
votes
1answer
62 views

Sequences where each number is a divisor of one less than the next

Let $N,k$ be fixed. Call a sequence of positive integers $a_1,a_2,\dots,a_k$ good if for each $i$, $a_i$ is a divisor of $a_{i-1}-1$. Consider the set $$S = \{a_i : a_1,\dots,a_k \text{ is a good ...
1
vote
0answers
73 views

Summation of no. of divisors

Let $d(n)$ = no. of divisors of $n$ and $(d(n))^2$ = square of no. of divisors of n. Let $$S(N) = \sum_{n=1}^{N} d(n)$$ and $$S_2(N) = \sum_{n=1}^{N}(d(n))^2$$ $$S(N) = ...