All Questions
Tagged with divisor-sum number-theory
364
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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
2
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1
answer
69
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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
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44
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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
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1
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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
$$\Delta_t(s)= - \sqrt{\frac{t}{s}}Y_1{(4\pi\sqrt{ts})}+ \sqrt{\frac{t}{...
- 756
2
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139
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Divisors sum and Bessel Function related sums
Discovered the following relation:
$$\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
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1
answer
103
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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
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1
answer
104
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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
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1
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66
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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
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Divisors Sum Related Interesting Approximate Relation
Working on Divisors Sum Efficient calulcation topic. Accidentaly discovered one interesting relation which is accurate up to $10^{17}$ order.
$$\sum_{i=1}^{\infty}{\frac{\sigma(i)}{e^{i}}}\approx\frac{...
- 1,818
2
votes
0
answers
62
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patterns in the abundancy index of integers
Let $\sigma(n)$ be the sum of all divisors (including 1 and $n$) of $n$, and define the abundancy index of $n$ as $I(n) = \sigma(n)/n $. For example: $I(6)= \frac{1+2+3+6}{6} = 1/1+1/2 +1/3 +1/6 = 2$. ...
- 702
2
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1
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62
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On the "efficiency" of digit-sum divisibility tricks in other bases (or about the growth rate of the number of divisors function).
Out of the different divisibility tricks there's a really simple rule that works for more than one divisor: The digit-sum. Specifically, if the sum of the digits of a number is a multiple of $1,3$ or $...
- 7,273
2
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1
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48
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Solutions of equations of form $\tau(n+a_i)=\tau(x+a_i)$ where $\tau$ is the number of divisors function and $a_i$ is a diverging series
Consider the function $$\tau(n)=\sum_{d|n}1$$ which gives the number of divisors of a number. The question is: How much information does $\tau(n)$ contain about $n$?
The answer is obviously: not very ...
- 103
2
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1
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55
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The number of positive divisors of a number that are not present in another number.
How many positive divisors are there of $30^{2024}$ which are not divisors of $20^{2021}$?
I have tried many ways to try to get a pattern for this problem but I can't. I know that $30$ has $8$ ...
- 21
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92
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Growth rate of sum of divisors cubed [closed]
I a trying to find a result similar to:
$$\limsup_{n \to \infty} \frac{\sigma_1(n)}{n \log \log (n)} = e^\gamma$$
(where $\sigma_1$ is the sum of divisors function) but regarding the growth rate of $\...
- 1,919
5
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2
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262
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Determine all positive integers $n$ such that: $n+d(n)+d(d(n))+\dotsb=2023$.
For a positive integer number $n>1$, we say that $d(n)$ is its superdivisor if $d(n)$ is the largest divisor of $n$ such that $d(n)<n$. Additionally, we define $d(0)=d(1)=0$. Determine all ...
- 418