Questions tagged [divisor-sum]
For questions on the divisor sum function and its generalizations.
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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 ...
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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 ...
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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 - ...
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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 ($p_1 < p_2 < \dots < p_m$) ...
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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 ...
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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 ...
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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
$$\Delta_t(s)= - \sqrt{\frac{t}{s}}Y_1{(4\pi\sqrt{ts})}+ \sqrt{\frac{t}{...
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Divisors sum and Bessel Function related sums
Discovered the following relation:
$$\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(...
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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 ...
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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 ...
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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
$$
\sigma_1 (z)...
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Efficient proof that a number is NOT a Zumkeller number?
The subset sum problem is known to be NP-complete , so in general there is no efficient method to decide it , in particular to prove a negative result.
This problem arises in the problem to decide ...
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Exercise 1 Section 9.2. Montgomery/Vaughan's Multiplicative NT
I am trying to show that for any integer $a$, $$e(a/q) =
\sum_{d|q, d|a} \dfrac{1}{ϕ(q/d)} \sum_{χ \ (mod \ q/d)} χ(a/d) τ(χ).$$ First I considered the case $(a,q)=1$ and the mentioned equality holds. ...
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Can we efficiently check whether a number is a Zumkeller number?
A positive integer $n$ is a Zumkeller number iff its divisors can be partitioned into two sets with equal sum.
If $\sigma(n)$ denotes the divisor-sum-function , this means that there are distinct ...
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Asymptotic for $\sum_{d\mid N}\frac{d^{2}}{\sigma\left(d\right)}$ [closed]
Let $N\in\mathbb{N}$. I'm looking for an asymptotic formula for $\sum_{d\mid N}\frac{d^{2}}{\sigma\left(d\right)}$ as $N\rightarrow\infty$. I tried to use known asymptotic formulas for similar ...
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Divisors Sum Related Interesting Approximate Relation
Working on Divisors Sum Efficient calulcation topic. Accidentaly discovered one interesting relation which is accurate up to $10^{17}$ order.
$$\sum_{i=1}^{\infty}{\frac{\sigma(i)}{e^{i}}}\approx\frac{...
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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: $I(6)= \frac{1+2+3+6}{6} = 1/1+1/2 +1/3 +1/6 = 2$. ...
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Proof for the sum and number of positive divisors for a positive integer $n$. [duplicate]
I know that the number of positive divisors of $n$ can be given by :
$\tau(n)$ = $(a_1+1)(a_2+1)\ldots(a_k+1)$ where $n = p_1^{a_1}p_2^{a_2}.... p_k^{a_k}$, where $p_1, p_2... p_k$ are the prime ...
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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 $...
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Upper bounds on the greatest common divisor of sums of geometric series
Let $S_1=\sum_{i=0}^{n} p^i = \frac{p^{n+1}-1}{p-1}$ and $S_2=\sum_{i=0}^{m} q^i = \frac{q^{n+1}-1}{q-1}$ be two sums of geometric series, and $\gcd\left(S_1,S_2\right)$ its greatest common divisor. ...
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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 ...
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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 $8$ ...
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Maxima of the Sum of Divisors Function and Upper Bound of a similar Ratio
If we define a function (aka Gronwall’s function) as: $$F(n)=\frac{\sigma(n)}{n \log \log n}$$
Then for $n>15$, it does have an upper bound. I want to know what's that specific upper bound is?
Also ...
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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 $\...
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Why must $n-\lfloor\frac n2\rfloor+\lfloor\frac n3\rfloor-\dotsb$ grow like $n\ln2$? [duplicate]
Let
$$a(n):=n-\left\lfloor\frac n2\right\rfloor+\left\lfloor\frac n3\right\rfloor-\left\lfloor\frac n4\right\rfloor+\left\lfloor\frac n5\right\rfloor-\dotsb.$$
Note that this is a finite sum. Naïvely,
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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 ...
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Square of prime numbers
This conjecture is inspired by the comment of @Eric Snyder: Prime numbers which end with 03, 23, 43, 63 or 83
$n$ is a natural number $>1$, $\varphi(n)$ denotes the Euler's totient function, $P_n$ ...
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Prime numbers which end with 03, 23, 43, 63 or 83
This is inspired from this post: Prime numbers which end with $19, 39, 59, 79$ or $99$
Here I found a new formula:
$n$ is a natural number $>1$, $\varphi(n)$ denotes the Euler's totient function, $...
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Can the sum of odd divisors of an integer exceed n [closed]
I know that the sum of divisors of an integer n can exceed it (abundant numbers) but can this occur when only considering odd divisors of n? Can the sum of all integers j : j|n, 2∤j be greater than n? ...
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Prime numbers which end with $59$ or $79$ [closed]
This is the weak conjecture of https://math.stackexchange.com/questions/4834936/prime-numbers-which-end-with-19-39-59-79-or-99.\
$\varphi(n)$ denotes the Euler’s totient function, $n$ denotes a ...
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