Questions tagged [divisor-counting-function]
For questions that involve the divisor counting function, also known as $\sigma_0$, $\tau$, or $d$.
278 questions
6
votes
2
answers
195
views
When does $a^n+1$ have $n$ divisors?
For positive integers $a$ and $n$, when does $a^n+1$ have $n$
divisors?
This was a natural question that popped up when I was investigating numbers of the form $a^n+1$, but surprisingly I can't seem ...
- 2,324
0
votes
2
answers
106
views
The number of divisors of $n$ and the sum of divisors of $n^2$?
Let $\tau(n)$ be the number of positive divisors of $n$, and let $\sigma(n)$ be the sum of its positive divisors.
I was playing around with these functions for small values of $n$ and noticed ...
- 19
1
vote
3
answers
163
views
Fraction of numbers with a number of divisors divisible by $3$
I think that the fraction of numbers with a number of divisors divisible by $3$ is $1-\frac{6}{\pi^2}\zeta(3)$.
To formally define what I mean by the fraction of numbers with a certain property, if $f(...
- 61
0
votes
2
answers
153
views
Ramanujan's theorems about the sums of powers of the number of divisors
This is from Hardy and Wright’s An Introduction to the Theory of Numbers, Section 18.2. "The average order of $d(n)$". Here, $d(n)$ denotes the number of divisors of $n$. The section states ...
- 1,563
2
votes
0
answers
267
views
Trying to understand a previously posted answer
I was given the following question :-
Let $\tau_k(n)$ count the number of ways of representing $n$ as the product of $k$ natural numbers. For $k\ge 2$, prove that
$$D_k(x) := \sum_{n \leq x} \tau_k(n)...
user1096489
0
votes
0
answers
94
views
Prove $\sum_{n \leq x} \tau_k(n) = xP_k(\log x) + O_{\varepsilon, k}\left (x ^{1 - \frac{1}{k}+\varepsilon}\right)$ [duplicate]
I was given the following question :-
Let $\tau_k(n)$ count the number of ways of representing $n$ as the product of $k$ natural numbers. For $k\ge 2$, prove that
$$D_k(x) := \sum_{n \leq x} \tau_k(n)...
user1096489
0
votes
2
answers
140
views
Number of divisor of $5^{7}\times 7^{10}$ of the form $3k+2$, $k\geq 0$
Reference Question:
The number of and sum of all divisor which are odd and are of the form $3k+2$ for the number $n=2^{7}\times 3^{2}\times 5^{1}\times 7^{1}$
My approach: Since number is odd and is ...
- 4,708
0
votes
2
answers
112
views
Find number of divisor of $n=3^{5}\cdot 5^{7}\cdot 7^{9}$ which are of the form $4k+1$, $n\geq 0$
Find number of divisor of $n=3^{5}\cdot 5^{7}\cdot 7^{9}$ which are of the form $4k+1$, $k\geq 0$
My solution: Total number of divisor of $n$ are $(5+1)(7+1)(9+1)=480$
All the divisor can be of the ...
- 4,708
3
votes
2
answers
177
views
Lower bound for $\frac{\sigma_1(n)}{\sigma_0(n)}$?
Notation: Here I take $\sigma_x(n)$ to mean $\displaystyle{\sum_{d|n}} d^{x}$.
Is there a good lower bound known for $\dfrac{\sigma_1(n)}{\sigma_0(n)}$?
I've seen this post on the average order of ...
- 199
3
votes
0
answers
48
views
estimating an elementary sum involving divisor function
Please guide me as to how to obtain the below bound and whether it is optimal.
Let a squarefree integer $N=\prod_{1 \leq i \leq m} p_i$ be a product of $m$ primes ($p_1 < p_2 < \dots < p_m$) ...
- 417
0
votes
0
answers
101
views
Highly composite numbers which are the middle of a twin prime
From the first $10\ 000$ highly composite numbers listed in OEIS , the following $20$ are the middle of a twin-prime that is we have a highly composite number $N$ such that both $N-1$ and $N+1$ are ...
- 87k
2
votes
1
answer
76
views
Solutions of equations of form $\tau(n+a_i)=\tau(x+a_i)$ where $\tau$ is the number of divisors function and $a_i$ is a diverging series
Consider the function $$\tau(n)=\sum_{d|n}1$$ which gives the number of divisors of a number. The question is: How much information does $\tau(n)$ contain about $n$?
The answer is obviously: not very ...
- 155
2
votes
1
answer
290
views
Prove that φ(n) + d(n) ≤ n + 1 [duplicate]
Prove that φ(n) + d(n) ≤ n + 1.
d(n) is the number of positive divisors of n.
φ(n) is the Euler's Totient Function.
Attempt：
For a prime number n, φ(n) = n - 1 (all numbers less than n are relatively ...
2
votes
0
answers
81
views
Numbers of divisors for a Mersenne number
Recently I encountered a problem:
If n is a positive integer, then is the number of divisors of $2^n - 1$ less or greater than the number of divisors of n?
I tried factoring and taking modulo n but ...
- 21
1
vote
2
answers
206
views
Call $n\in\Bbb N$ "balanced" if the sum of its digits equals the count of its divisors. How many "balanced" numbers are there up to $m$?
I recently stumbled across a problem about numbers' divisor count (more specifically, how many positive integers are equal to the square of their divisor count - answer was 2: they are 1 and 9).
But I ...
7
votes
0
answers
136
views
Asymptotic on $\sum_{n<x} 1/d_{\alpha}(n)$, $d_\alpha$ is the general divisor function.
Let $d(n)$ be the divisor function, that is, $$d(n)=\sum_{1\leq k\leq n, \,k|n} 1.$$ Or equivalently, $d(n)$ can be defined as the coefficients of $\zeta^2(s)$: $$\zeta^2(s)=\sum_{n\geq1} d(n)n^{-s}.$$...
- 2,033
6
votes
1
answer
332
views
Optimal bounds for the product of the divisor function $d(n)$ in short intervals
Let $d(n)$ denote the number of divisors of a positive integer $n$.
It is pretty obvious that $d(n) \ge 2$ for any given number $n \ge 2$, since every number is divisible by $1$ and itself. $2$ is ...
- 1,391
2
votes
1
answer
142
views
The main term when counting $d(n)$ in arithmetic progressions
For $q,a\in \mathbb N$ write $d=(q,a)$. Why might be
$$\sum _{h|q}\frac {c_h(a)\log (d/h)}{h}=-\frac {q'}{\phi (q')}\sum _{h|q}\frac {c_h(a)}{h}\sum _{h'|q'}\frac {\mu (h')\log h'}{h'}+\sum _{h|d}\...
- 1,821
4
votes
2
answers
980
views
How did Ramanujan came up with this?
The following is a picture of equation from Ramanujan's lost notebook. In this page, Ramanujan gives a closed form for,
$$\sum_{n\geq 1}\sigma_{s}(n)x^{n}$$
In an attempt initially it's claimed that,
$...
- 1,620
0
votes
1
answer
130
views
What is the Dirichlet Convolution of the identity function with itself?
If you have two identity functions, then $f(d) * g(n/d)$ would be just $dn/d = n$. Since we have an $n$ added for each divisor of $n$, would the resulting function just be $n$ times the number of ...
1
vote
1
answer
479
views
Finding a formula for $\sum_{d|n} \tau(d)$, where $\tau(d)$ is the number of divisors of $d$.
I am currently in the middle of the following exercise:
Exercise. Compute $$ \sum_{d|n} \left(\sigma(d)\mu\left(\frac{n}{d}\right)+\tau(d)\right),$$
where $\sigma$ is the function that corresponds to ...
- 1,289
1
vote
1
answer
138
views
Hand calculation of divisor summatory function
Maybe this question is stupid, but there was a problem in a math competition (not even in the highest stage) in my county which asked to
Find $ \sum_{n\leq390} d(n)$, where $d(n)$ is the number of ...
- 47
1
vote
0
answers
64
views
How can we show this relationship between the recursive sum of divisors function and Figurate Number Polynomial on Primes?
Let us define the following recursive function involving the sum of divisors function $\sigma(n)$:
\begin{array}{ l }
r(n,1)=\sigma(n) \\
r(n,2)=\sum_{d|n}r(d,1) \\
r(n,3)=\sum_{d|n}r(d,2) \\
\...
- 1,678
5
votes
1
answer
142
views
How can we show this relationship between the sum of divisors function and the sum $p^{m}+2p^{m-1}+3p^{m-2}+\ldots+(m+1)$?
The sum of divisors function is commonly denoted by $\sigma(n)$. Now let us introduce a recursive definition of divisor functions:
$r_{n,1}=\sigma(n)$
$r_{n,2}=\sum_{d|n}r_{d,1}$
$r_{n,3}=\sum_{d|n}...
- 1,678
0
votes
1
answer
116
views
Question involving series of divisor function and Euler function
We shall prove that $\sum_{n=1}^{+\infty} \frac{d(n)}{2^n}=\sum_{n=1}^{+\infty} \frac{1}{\phi(2^{n+1}-1)}$, where d(n) the divisor function.
I was thinking of making use of the fact that d(n) is ...
