Questions tagged [divisor-sum]

For questions on the divisor sum function and its generalizations.

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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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Repitition in the pattern of $\tau(n)$ [closed]

I am interested in the function $\tau(n)=\sum_{d\mid n}1$ which gives the number of positive divisors of $n$. I have the following two questions related to this: Does there exist a $k$ such that $$\...
xyz1234's user avatar
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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 divisors and 20 has ...
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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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
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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 ...
Kokos's user avatar
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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, $P_n$ ...
Rédoane Daoudi's user avatar
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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, $...
Rédoane Daoudi's user avatar
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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? ...
dean's user avatar
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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 ...
Rédoane Daoudi's user avatar
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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)...
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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
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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(...
Daniel Donnelly's user avatar
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About the sum of divisors of $2n$ that do not divide $n$

Taking $\sigma(n)$ as the sum of divisors of $n$, it is possible to write the sum of divisors of $2n$ that do not divide $n$ as $\sigma(2n) - \sigma(n)$. Now, I found this relation by working with a ...
Matheus Silva Carvalho de Oliv's user avatar
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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
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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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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 ...
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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
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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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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
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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
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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
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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
422 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
169 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
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139 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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On the divisibility constraint $\sigma(q^k)/2 \nmid n$ and the Descartes spoof

Let $N = q^k n^2$ be a hypothetical 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 $...
Jose Arnaldo Bebita Dris's user avatar
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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
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On the equations $D(q^k)D(n^2)=2s(q^k)s(n^2)$ and $\sigma(q^k)\sigma(n^2)=2 q^k n^2$, where $q^k n^2$ is an odd perfect number with special prime $q$

MOTIVATION The topic of odd perfect numbers likely needs no introduction. In what follows, we denote the classical sum of divisors of the positive integer $x$ by $\sigma(x)=\sigma_1(x)$, the ...
Jose Arnaldo Bebita Dris's user avatar
1 vote
1 answer
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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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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 ...
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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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Is there an infinite composite numbers $n$ such that $\sigma(\sigma(n)+n) = 2 \cdot \sigma(n)$

For a positive integer $n$, let $\sigma(n)$ denote the sum of the positive divisors of $n$. Now, suppose that $n$ is a composite number. Is there an infinitely many $n$ composite such that $\sigma(\...
Craw Craw's user avatar
4 votes
1 answer
273 views

Determining whether $\sigma(q^k)/2$ is squarefree, where $q^k n^2$ is an odd perfect number with special prime $q$

Preamble: The present inquiry is an offshoot of What are the remaining cases to consider for this problem, specifically all the possible premises for $i(q)$?. MOTIVATION Denote the classical sum of ...
Jose Arnaldo Bebita Dris's user avatar
1 vote
1 answer
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Is this disproof for the Descartes-Frenicle-Sorli Conjecture that $k=1$, if $p^k m^2$ is an odd perfect number, valid?

Let $N = p^k m^2$ be an odd perfect number with special prime $p$ satisfying $p \equiv k \equiv 1 \pmod 4$ and $\gcd(p,m)=1$. It is known that $$D(p^k)D(m^2)=2s(p^k)s(m^2) \tag{0}$$ where $D(x)=2x-\...
Jose Arnaldo Bebita Dris's user avatar
2 votes
2 answers
148 views

Divisors count of $3^n\pm1$

During the investigation of $3^n\pm1$ visually saw that the divisors count of it are sums of $2^m$. Especially if we take $n=p_1p_2$ ($p_i$ is prime) then at least for $n=p_1p_2<130$ it is either ...
Gevorg Hmayakyan's user avatar
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1 answer
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If $p^k m^2$ is an odd perfect number, then $D(p^k)/s(p^k)$ is in lowest terms. Does this contradict $D(p^k)D(m^2)=2s(p^k)s(m^2)$?

In what follows, denote the classical sum of divisors of the positive integer $x$ by $\sigma(x)=\sigma_1(x)$. If $n$ is odd and $\sigma(n)=2n$, then $n$ is called an odd perfect number. Euler showed ...
Jose Arnaldo Bebita Dris's user avatar
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Is there an analytical solution to the inequality $\frac{1 + k({p}^{(k+1)/2)})}{p^k} < \frac{p}{p - 1}$ if one were to bound $k$ in terms of $p$?

My question is as is in the title: Is there an analytical solution to the inequality $$\frac{1 + k({p}^{(k+1)/2)})}{p^k} < \frac{p}{p - 1}$$ if one were to bound $k$ in terms of $p$? Here, $p \...
Jose Arnaldo Bebita Dris's user avatar
2 votes
0 answers
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Is there an example for every given exponent?

Inspired by this question , where it is asked for positive integers with the property $$2n-\sigma(n)\mid \sigma(n)-n$$ which is equivalent to $$2n-\sigma(n)\mid n$$ The author also demands $2n-\sigma(...
Peter's user avatar
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Why is it that, if there are no odd perfect numbers, then there are no other $3$-perfect numbers, apart from the six known, as of the year $1643$?

Let $\sigma(x)=\sigma_1(x)$ denote the classical sum of divisors of the positive integer $x$. A number $N$ is said to be $k$-perfect if $\sigma(N)=kN$ where $k$ is a positive integer. The number $1$ ...
Jose Arnaldo Bebita Dris's user avatar
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0 answers
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Sum of Factors Question [duplicate]

Let $d_1,d_2,\ldots,d_k,$ be all the factors of a positive integer $n,$ including $1,$ and $n.$ Suppose $d_1+d_2+\ldots+d_k=72.$ Then, find the value of $\frac{1}{d_1}+\frac{1}{d_2}+\ldots+\frac{1}{...
aqualubix's user avatar
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1 answer
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Euler product involving divisor function

Take $k,N\in \mathbb N$ and $s\in \mathbb C$ with real part $\sigma \in [1-\delta ,1]$ for some small fixed $\delta $. In its simplest form my question is how do I sum $$\sum _{l\geq 0}\frac {d_k(p^{...
tomos's user avatar
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Define $\partial(n\mid x) = \partial(q_0 \cdots q_i \mid x) = \sum_{j=0}^i (-1)^i q_j (\frac{n}{q_j} \mid x)$. What does homology measure?

Let $R$ be a commutative ring with $1$ and let $M = \{ f : \Bbb{N} \to R \}$ be the $R$-module of arithmetic functions into $R$. A basis for $M$ is $(d \mid \cdot) : d \in \Bbb{N}$ where $(d\mid n) = ...
Daniel Donnelly's user avatar
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1 answer
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If $p$ is a prime number and $k$ is a positive integer, is it true that $\sigma_1(p^k) > 1 + k (\sqrt{p})^{1+k}$?

Denote the classical sum of divisors of the positive integer $x$ by $\sigma(x)=\sigma_1(x)$. Here is my initial question: If $p$ is a prime number and $k$ is a positive integer, is it true that $$\...
Jose Arnaldo Bebita Dris's user avatar
1 vote
1 answer
135 views

If $p^k m^2$ is an odd perfect number with special prime $p$ and $p = k$, then $\sigma(p^k)/2$ is not squarefree.

While researching the topic of odd perfect numbers, we came across the following implication, which we currently do not know how to prove: CONJECTURE: If $p^k m^2$ is an odd perfect number with ...
Jose Arnaldo Bebita Dris's user avatar
3 votes
1 answer
95 views

The mean square of $d_k(n)$

Let $d_2(n)=d(n)$ be the divisor function, and let $$d_k(n)=\sum_{d_1\cdots d_k= n}1=\sum_{m\cdot l= n}d_{k-1}(m).$$ Can anyone point me to a reference to the size of the error term when approximating ...
tomos's user avatar
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On the prime factorization of $n$ and the quantity $J = \frac{n}{\gcd(n,\sigma(q^k)/2)}$, where $q^k n^2$ is an odd perfect number

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
-1 votes
1 answer
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On a consequence of $G \mid I \iff \gcd(G, I) = G$ (Re: Odd Perfect Numbers and GCDs)

Let $N = q^k n^2$ be an odd perfect number given in the so-called Eulerian form, where $q$ is the special prime satisfying $q \equiv k \equiv 1 \pmod 4$ and $\gcd(q,n)=1$. Denote the classical sum of ...
Jose Arnaldo Bebita Dris's user avatar

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