Questions tagged [divisor-sum]
For questions on the divisor sum function and its generalizations.
852
questions
0
votes
0
answers
23
views
Does the following GCD divisibility constraint imply that $\sigma(m^2)/p^k \mid m$, if $p^k m^2$ is an odd perfect number with special prime $p$?
The topic of odd perfect numbers likely needs no introduction.
In what follows, denote the classical sum of divisors of the positive integer $x$ by $$\sigma(x)=\sigma_1(x).$$
Let $p^k m^2$ be an odd ...
- 8,010
0
votes
0
answers
25
views
Improving $I(m^2)/I(m) < 2^{\log(13/12)/\log(13/9)}$ where $p^k m^2$ is an odd perfect number with special prime $p$
In what follows, let $I(x)=\sigma(x)/x$ denote the abundancy index of the positive integer $x$, where $\sigma(x)=\sigma_1(x)$ is the classical sum of divisors of $x$.
The following is an attempt to ...
- 8,010
0
votes
0
answers
51
views
Divisors of $x^2-1$ in Brocard's Problem
In this post, I was curious if the divisor bound could be improved for the product of two consecutive even numbers. It seems most likely it cannot by much. How could the upper bound of $\sigma_0(x^2 - ...
- 33
3
votes
0
answers
32
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
2
votes
1
answer
69
views
Generalized "perfect numbers" using different n,k values of divisorSum[n, k]
Using the divisor_sigma[n, k] function from the python sympy library where n is the positive integer which is having its divisors added and k is the power each factor is raised to, I was looking for ...
- 31
0
votes
0
answers
44
views
Why are their common ratios of integers to the sums of their proper divisors?
I was playing around on Desmos with a function that computed the sums of proper divisors of an integer and found an interesting pattern regarding the "slopes" of the graph:
Graph of integers ...
- 9
2
votes
1
answer
97
views
these pde's and the Dirichlet divsor problem
I noticed that
$$t^2 \frac{\partial^3}{\partial t^3}\Delta_t(s)+s^2 \frac{\partial}{\partial s} \Delta_t(s)=0 $$
is satisfied by
$$\Delta_t(s)= - \sqrt{\frac{t}{s}}Y_1{(4\pi\sqrt{ts})}+ \sqrt{\frac{t}{...
- 756
2
votes
0
answers
139
views
Divisors sum and Bessel Function related sums
Discovered the following relation:
$$\sum _{k=1}^{\infty } \sigma (k) \left(K_2\left(4 \pi \sqrt{k+y} \sqrt{y}\right)-K_0\left(4 \pi \sqrt{k+y} \sqrt{y}\right)\right)=\frac{\pi K_1(4 \pi y)-3 K_0(...
- 1,818
1
vote
1
answer
103
views
Inverse Mellin transform of $\Delta(t):=\sum_{s=1}^\infty d(s)\sqrt{\frac{t}{s}}M_1(4\pi \sqrt{ts})$
Define
$$ \Delta(t):=\sum_{s=1}^\infty d(s)\sqrt{\frac{t}{s}}M_1(4\pi \sqrt{ts}) $$
where $M_1(z)=-Y_1{(z)}-\frac{2}{\pi}K_1(z)$ and $d(s)$ is the divisor function.
What is the inverse Mellin ...
- 756
3
votes
1
answer
104
views
Approximation of $\sigma(n)$ sum.
Investigating:
$$\epsilon(n)=\frac{(\pi -3) e^{2 \pi n}}{24 \pi }-\sum _{k=1}^n \sigma(k) e^{2 \pi (n-k)}$$
where $\sigma(n)$ is a divisors sum of $n$.
Using long calculations (can not share here ...
- 1,818
0
votes
0
answers
19
views
How do I use the gaussian divisors formula?
For an integer z,
$$
z = \epsilon \prod_i p_i^{k_i},
$$
where $\epsilon$ is and a unit and every $p_i$ is a Gaussian prime in the first quadrant then the sum of the Gaussian divisors is
$$
\sigma_1 (z)...
2
votes
0
answers
63
views
Efficient proof that a number is NOT a Zumkeller number?
The subset sum problem is known to be NP-complete , so in general there is no efficient method to decide it , in particular to prove a negative result.
This problem arises in the problem to decide ...
- 85.1k
0
votes
2
answers
70
views
Exercise 1 Section 9.2. Montgomery/Vaughan's Multiplicative NT
I am trying to show that for any integer $a$, $$e(a/q) =
\sum_{d|q, d|a} \dfrac{1}{ϕ(q/d)} \sum_{χ \ (mod \ q/d)} χ(a/d) τ(χ).$$ First I considered the case $(a,q)=1$ and the mentioned equality holds. ...
- 281
5
votes
0
answers
71
views
Can we efficiently check whether a number is a Zumkeller number?
A positive integer $n$ is a Zumkeller number iff its divisors can be partitioned into two sets with equal sum.
If $\sigma(n)$ denotes the divisor-sum-function , this means that there are distinct ...
- 85.1k
0
votes
1
answer
66
views
Asymptotic for $\sum_{d\mid N}\frac{d^{2}}{\sigma\left(d\right)}$ [closed]
Let $N\in\mathbb{N}$. I'm looking for an asymptotic formula for $\sum_{d\mid N}\frac{d^{2}}{\sigma\left(d\right)}$ as $N\rightarrow\infty$. I tried to use known asymptotic formulas for similar ...
- 49