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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 ...
Jose Arnaldo Bebita Dris's user avatar
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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 ...
Jose Arnaldo Bebita Dris's user avatar
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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 - ...
PiMaster's user avatar
3 votes
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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 ($p_1 < p_2 < \dots < p_m$) ...
user166305's user avatar
2 votes
1 answer
64 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 ...
unnamed's user avatar
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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 ...
IvesM's user avatar
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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}{...
zeta space's user avatar
2 votes
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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(...
Gevorg Hmayakyan's user avatar
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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 ...
zeta space's user avatar
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 ...
Gevorg Hmayakyan's user avatar
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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)...
nullmatrix's user avatar
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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 ...
Peter's user avatar
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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. ...
Ali's user avatar
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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 ...
Peter's user avatar
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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 ...
user23571119's user avatar
6 votes
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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{...
Gevorg Hmayakyan's user avatar
2 votes
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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: $I(6)= \frac{1+2+3+6}{6} = 1/1+1/2 +1/3 +1/6 = 2$. ...
AndroidBeginner's user avatar
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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 ...
1025's user avatar
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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 $...
Robert Lee's user avatar
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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. ...
Juan Moreno's user avatar
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2 votes
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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 ...
xyz1234's user avatar
  • 103
2 votes
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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 $8$ ...
Ahmed Amir's user avatar
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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 ...
Ok-Virus2237's user avatar
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1 answer
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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 $\...
user3141592's user avatar
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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, ...
Akiva Weinberger's user avatar
5 votes
2 answers
261 views

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 ...
Kokos's user avatar
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3 votes
2 answers
491 views

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, $P_n$ ...
Craw Craw's user avatar
3 votes
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158 views

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, $...
Craw Craw's user avatar
1 vote
0 answers
52 views

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? ...
dean's user avatar
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1 answer
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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 ...
Craw Craw's user avatar
6 votes
1 answer
257 views

A question about prime numbers, totient function $ \phi(n) $ and sum of divisors function $ \sigma(n) $

I noticed something interesting with the totient function $ \phi(n) $ and sum of divisors function $ \sigma(n) $ when $n > 1$. It seems than : $ \sigma(4n^2-1) \equiv 0 \pmod{\phi(2n^2)}$ only if ...
Aurel-BG's user avatar
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Is there any square harmonic divisor number greater than $1$?

A harmonic divisor number or Ore number is a positive integer whose harmonic mean of its divisors is an integer. In other words, $n$ is a harmonic divisor number if and only if $\dfrac{nd(n)}{\sigma(n)...
Jianing Song's user avatar
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1 vote
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Extraordinary Numbers

Can you please explain what are Extraordinary Numbers in detail? At the same time, I would also like to confirm whether the equivalent problem of Riemann Hypothesis mentioned here is correct (like it'...
Ok-Virus2237's user avatar
1 vote
1 answer
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$\sum_{k = 1}^{\infty} k\lfloor\frac{n}{k} \rfloor = 1 + \sum_{k = 1}^n \sigma_1(n)$

For any $f: \Bbb{N} \to \Bbb{Z}$ there exists a unique transformed function $F:\Bbb{N} \to \Bbb{Z}$ such that: $$ f(n) = \sum_{k = 1}^{\infty}F_k\lfloor\frac{n}{k}\rfloor $$ For example, set $F_1 = f(...
SeekingAMathGeekGirlfriend's user avatar
1 vote
1 answer
83 views

Prove that there are infinitely many natural number such that $σ(n)>100n$

The problem is as follows: Prove that there are infinitely many natural numbers such that $σ(n)>100n$. $σ(n)$ is the sum of all natural divisors of $n$ (e.g. $σ(6)=1+2+3+6=12$). I have come to the ...
Mathology's user avatar
1 vote
2 answers
156 views

Finding natural numbers with $12$ divisors $1=d_1<d_2<\cdots<d_{12}=n$, such that the divisor with the index $d_4$ is equal to $1+(d_1+d_2+d_4)d_8$.

Find the natural number(s) n with $12$ divisors $1=d_1<d_2<...<d_{12}=n$ such that the divisor with the index $d_4$, i.e, $d_{d_4}$ is equal to $1+(d_1+d_2+d_4)d_8$. My work: $$\begin{align} ...
Rijhi's user avatar
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1 answer
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Question on an equation involving sum of a function over divisors. [closed]

I have a simple question regarding a particular form of a sum and I was hoping someone could provide some insights or guidance. I was wondering if there was any other way to express the following sum ...
AmB's user avatar
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1 answer
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Equality of two sums involving hecke eigenvalues in a paper of Luo and Sarnak

I am reading the paper Mass Equidistribution for Hecke Eigenforms by Luo and Sarnak. In the paper there is the following equality: By the multiplicativity of Hecke eigenvalues, we have $$ \sum_{r\geq ...
Steven Creech's user avatar
1 vote
1 answer
79 views

Continued aliquot sums

What happens if one takes the aliquot sum of an integer and then repeats the process so that one takes the aliquot sum of all of those factors that were not reduced to the number 1 on the previous ...
Robert J. McGehee's user avatar
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111 views

Limit of convolution sums of divisor functions

In this paper, Ramanujan studies the convolution sum of divisor functions, which he denotes as $$\sum_{r,s}(n) := \sum_{m = 0}^n \sigma_r(m) \sigma_s(n-m),$$ where above, he defines $\sigma_s(0) = \...
Mary_Smith's user avatar
1 vote
1 answer
144 views

On a conjecture involving multiplicative functions and the integers $1836$ and $137$

We denote the Euler's totient as $\varphi(x)$, the Dedekind psi function as $\psi(x)$ and the sum of divisors function as $\sigma(x)$. Are well-known arithmetic functions, see the corresponding ...
user759001's user avatar
4 votes
1 answer
107 views

On a conjecture involving multiplicative functions and the integers $1836$ and $136$

We denote the Euler's totient as $\varphi(x)$, the Dedekind psi function as $\psi(x)$ and the sum of divisors function as $\sigma(x)$. Are well-known arithmetic functions, see Wikipedia. I would like ...
user759001's user avatar
1 vote
1 answer
320 views

Who discovered the largest known $3$-perfect number in $1643$?

Multiperfect numbers probably need no introduction. (These numbers are defined in Wikipedia and MathWorld.) I need the answer to the following question as additional context for a research article ...
Jose Arnaldo Bebita Dris's user avatar
7 votes
2 answers
447 views

Is this a new representation of (some) Bernoulli numbers?

Let $\operatorname{B}(n)$ denote the Bernoulli numbers and $\operatorname{b}(n) = \operatorname{B}(n)/n$ with $b(0)=1$ the divided Bernoulli numbers. Also let $\sigma_{k}(n)= \sum_{d \mid n} d^k$ ...
Peter Luschny's user avatar
2 votes
2 answers
183 views

Does an odd perfect number have a divisor (other than $1$) which must necessarily be almost perfect?

Let $\sigma(x)=\sigma_1(x)$ be the classical sum of divisors of the positive integer $x$. Denote the aliquot sum of $x$ by $s(x)=\sigma(x)-x$ and the deficiency of $x$ by $d(x)=2x-\sigma(x)$. ...
Jose Arnaldo Bebita Dris's user avatar
1 vote
0 answers
141 views

Proving $n \mid \sigma(n^2)$

Let $N = q^k n^2$ be an odd perfect number with special prime $q$ satisfying $q \equiv k \equiv 1 \pmod 4$ and $\gcd(q,n)=1$. Denote the classical sum of divisors of the positive integer $x$ by $\...
Jose Arnaldo Bebita Dris's user avatar
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0 answers
42 views

Is this proof for the divisibility constraint $\sigma(q^k)/2 \mid n$ correct, where $q^k n^2$ is an odd perfect number with special prime $q$?

Let $N = q^k n^2$ be an odd perfect number given in Eulerian form, i.e. $q$ is the special prime satisfying $q \equiv k \equiv 1 \pmod 4$ and $\gcd(q,n)=1$. Define the GCDs $$G = \gcd(\sigma(q^k),\...
Jose Arnaldo Bebita Dris's user avatar
1 vote
1 answer
86 views

Solution of $\sigma(\sigma(m)+m)=2\sigma(m)$ with $\omega(m)>8$?

This question is related to this one. $\sigma(n)$ is the divisor-sum function (the sum of the positive divisors of $n$) and $\omega(n)$ is the number of distinct prime factors of $n$. The object is ...
Peter's user avatar
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1 vote
1 answer
99 views

Lower bound for divisor counting function

Let, $$\tau(n)=\sum_{d|n}1$$ Be the divisors counting function. Then is it true that, There exists infinitely many $n$ satisfying, $$\tau(n)>\left(\ln(n)\right)^{a}$$ Where $a\in[1,\infty)$? My ...
RAHUL 's user avatar
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1 vote
0 answers
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Is $n=2$ the only even solution of $\sigma(\sigma(n)+n)=2\sigma(n)$?

Inspired by this question. For positive integer $m$ , let $\sigma(m)$ be the divisor-sum function. Let $S$ be the set of positive integers $n$ satisfying $$\sigma(\sigma(n)+n)=2\sigma(n)$$ In the ...
Peter's user avatar
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