Questions tagged [divisors-multiples]

For questions on divisors and multiples, mainly but not exclusively of integers, and related and derived notions such as sums of divisors, perfect numbers and so on.

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A definition related to pseudoprimes and the Dedekind psi function

In this post we consider that $\psi(k)$ denotes the Dedekind psi function. Wikipedia has an artcle dedicated to this arithmetic function Dedekind psi function defined for a positive integers $m>1$ ...
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1 answer
214 views

Around the equation $\sigma\left(\square\right)=\text{prime}$: counterexamples or a proof for some of these conjectures

For integers $A,B\geq 1$ we define the difference $\sigma(A)\sigma(B)-\sigma(AB)$, denoting it as $[A,B]$, where $\sigma(n)=\sum_{1\leq d\mid n}d$ denotes the sum of divisors function. It is possible ...
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2 votes
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Analogue of Fermat's little theorem for Bernoulli numbers

Is the following analogue of Fermat's Little Theorem for Bernoulli numbers true? Let $D_{2n}$ be the denominator of $\frac{B_{2n}}{4n}$ where $B_n$ is the $n$-th Bernoulli number. If $\gcd(a, D_{2n}) ...
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How to compute/estimate the least $k$ such that there exist $n$ consecutive integers each having a prime factor $\le k$?

Let $a_n$ be the least integer $k$ such that there exist $n$ consecutive integers each with a prime factor $\le k$. For example, $a_{13} \le 11$ because the 13 consecutive integers $114,115,\ldots,126$...
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Does this Theorem 2 from Dandapat et al. imply that $\gcd(\sigma(p^k),\sigma(a^2)) > 1$?

Write the odd perfect number $m=p^k a^2$ as a product of primes $$m = p^k {p_1}^{2a_1} \cdots {p_v}^{2a_v}.$$ (Note that it is known that $v \geq 9$ by work of Nielsen.) Let $N(m)$ be the number of ...
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3 votes
2 answers
208 views

Help with R. Ryan's "A simpler dense proof regarding the abundancy Index."

I'm reading Richard Ryan's article "A simpler dense proof regarding the abundancy index" and got stuck in his proof for Theorem 2. The Theorem is stated as follows: Suppose we have a ...
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Estermann's argument for the binary additive divisor problem

In the paper https://eudml.org/doc/149759 an estimate for the binary additive divisor problem is given with a power saving. I don't get the main bit of the argument - I'm obviously missing something. ...
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The binary additive divisor problem in arithmetic progressions

I find quite a few results about the binary additive divisor problem, that is evaluating \[ \sum _{n\leq x}d(n)d(n+h)\] for certain ranges of $h$. Are there any known results about the same count ...
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Estimating from below positive moments of "clipped" divisor function on subsets of $\{1,2,\ldots,x\}$ with positive density

The question here about estimating positive moments of the divisor function on sets of nonzero density $A\subset \{1,2,\ldots,x\}$ was answered giving $$ S_a(x):=\sum_{n \in A} d(n)^a \geq |A|(\ln x)^{...
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Analogue of the second Hardy-Littlewood conjecture for numbers of divisors?

Let $f(n)$ denote the proposition "There exists some $k>1$ such that $$ \sum_{m=k}^{k+n-1}\tau(m) < \sum_{m=1}^n\tau(m) $$ where $\tau(m)$ is the number of the divisors of $m$." (This ...
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About inequalities that involve the sum of divisors, the Euler's totient and the aliquot part $\sigma(n)-n$

In this post, for integers $n\geq 1$, I denote the sum of divisors $\sum_{1\leq d\mid n}d$ as $\sigma(n)$ and the Euler's totient function as $\varphi(n)$. It's easy to check* that if we assume that $...
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3 answers
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On odd perfect numbers $p^k m^2$ with special prime $p$ satisfying $m^2 - p^k = 2^r t$ - Part II

(Preamble: We have asked this same question in MSE two weeks ago, without getting any answers. We have therefore cross-posted it to MO, hoping that it gets answered here.) The topic of odd perfect ...
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2 votes
1 answer
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Does $n \mid \sigma(n^2)$, if $q^k n^2$ is an odd perfect number?

Let $\sigma(x)=\sigma_1(x)$ be the classical sum of divisors of the positive integer $x$. It is known that $$\gcd(\sigma(q^k),\sigma(n^2))=\frac{\bigg(\gcd(n,\sigma(n^2))\bigg)^2}{\gcd(n^2,\sigma(n^2))...
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On odd perfect numbers and a GCD - Part III

Let $m = q^k n^2$ be an odd perfect number with special prime $q$ satisfying $q \equiv k \equiv 1 \pmod 4$ and $\gcd(q,n)=1$. It is known that $$\gcd(\sigma(q^k),\sigma(n^2)) = \frac{(\gcd(n,\sigma(n^...
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A diophantine equation involving partial sums of exponentials similar than the equation in Fermat's Last Theorem

I'm curious about the following diophantine equation from my invention: I don't know if this is in the literature, I wrote it using creativity in an attempt to write a variant of the equation in ...
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2 votes
1 answer
209 views

Behavior of biggest prime divisor of $n$ as $n$ grows large

Let $P\subseteq \mathbb{N}$ be the set of primes, and for any integer $n>1$ let $L(n) = \max\{p \in P: p \mid n\}$ be the largest prime divisor of $n$. Moreover, for $n \in \mathbb{N}$ with $n>1$...
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14 votes
2 answers
727 views

How many divisors of $n$ are below $n^{1/3}$?

I am trying to bound a function that includes $\sum\limits_{\substack{d < n^{1/3} \\ d \mid n}} 1$. Is there an upper bound known for this sum, either in general or in terms of $\sum\limits_{\...
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1 answer
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Proportionality constant in Montgomery-Vaughan Theorem 7.20

In Multiplicative Number Theory - Vol. I by Montgomery and Vaughan the following result is proved. Theorem 7.20 Let $A(x,r)$ denote the number of $n\leq x$ such that $\Omega(n)\leq r \log \log x,$ and ...
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A modern reference for the Piltz divisor problem

apparently, the Dirichlet hyperbola method is no longer up-to-date, and instead Voronoi's identity is used in order to establish good bounds on the Dirichlet divisor problem. The same applies to the ...
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On $q$, $k$ modulo $3$ and the residue of $\sigma(n^2)$ modulo $x$ when $q^k n^2$ is an odd perfect number with special prime $q$

Chen and Luo (now published) proved in Theorem 3.3, page 7 that if $m = q^k n^2$ is an odd perfect number with special prime $q$, then we have the biconditionals $$\sigma(n^2) \equiv 1 \pmod 4 \iff q \...
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Lower bound on a Truncated Divisor Sum

Let $d(n)$ be the number of divisors function, i.e., $d(n)=\sum_{k\mid n} 1$ of the positive integer $n$. I am interested in estimating, the following sum $$ A(a,x)=\sum_{n\leq x} \min[ d(n), M]^a $$ ...
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4 votes
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107 views

Factors of polynomials of bounded height

Let $f(x)=a_nx^n+\cdots+a_0 \in \mathbb{Z}[x]$ be an integer polynomial in one variable. Recall that the height $H(f):=\textrm{max}\,|a_n|$ is the largest coefficient. Consider the set of polynomials ...
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1 answer
316 views

On odd perfect numbers $q^k n^2$ satisfying $n^2 - q^k = 2^r t$

Let $N = q^k n^2$ be an odd perfect number with special prime $q$, satisfying $$n^2 - q^k = 2^r t$$ where $r \geq 2$ and $\gcd(2,t)=1$. We could prove that: (1) $2^r t > 2n$. (We can modestly ...
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2 answers
357 views

Finding all proper divisors of $a_3z^3 +a_2z^2 +a_1z+1$ of the form $xz+1$

Let $n=a_3z^3+a_2z^2+a_1z+1$ where $a_1<z, \ a_2<z, \ 1 \le a_3<z, z>1$ are non negative integers. To obtain proper divisors of $n$ of the form $xz+1$, one may perform trial divisions $xz+...
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Have any proposals been advanced for the analytic continuation of the divisor function?

While I was working on the evaluation of a certain series, the following limit came up: \begin{align} \lim_{n \to 1} \frac{d(n)-1}{n(n-1)} &= \lim_{n \to 1} \frac{d'(n)}{2n-1} \\ &= d'(1) .\...
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6 votes
0 answers
196 views

Growth rate of signed sum of $N \sigma_0(n)-\sigma_1(n)$

Let $\sigma_k(n)=\sum_{d|n} d^k,$ for a positive integer $n$ and $k\geq 0$. A lot is known about the averages for the functions $\sigma_k(n),$ such as the estimates $$ \sum_{n\leq x} \sigma_0(n)=x \...
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4 votes
1 answer
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Can the Lagarias inequality be written as a "kernel inequality"?

The Lagarias inequality, which is equivalent to the Riemann hypothesis, is: $$\sigma(n) \le H_n + \exp(H_n) \log(H_n) =:L(n)$$ for all natural numbers $n$, where $\sigma=$ sum of divisors, $H_n=n$-th ...
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1 vote
2 answers
342 views

Improving the lower bound $I(n^2) > \frac{2(q-1)}{q}$ when $q^k n^2$ 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 $\gcd(q,n)=1$. Define the abundancy index $$I(x)=\frac{\sigma(x)}{x}$$ where $\sigma(x)$ ...
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1 vote
1 answer
205 views

Moments of number of interval restricted divisors

I have previously asked the question A truncated divisor function sum where the sum $$ S_f(x)=\sum_{n\leq x} \min\{f(x),d(n)\}\quad (1) $$ was of interest, and it was answered satisfactorily. Here, I ...
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3 votes
0 answers
150 views

The kronecker symbol and factorization of $n=\frac{B^N-1}{B-1}$

Let $n=\frac{B^N-1}{B-1}$. Assume $n$ is congruent to 3 modulo 4. We have the following: If $N$ is 1 modulo 4, then $N$ is quadratic residue modulo $n$ and $-N$ is quadratic non-residue. The square ...
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3 votes
1 answer
366 views

Is there a Riemann Hypothesis criterion utilizing sum of squares of divisors?

Robin's inequality $$\sigma_1(n)<e^\gamma n\log\log n$$ at integers $n>5040$ provides necessary and sufficient condition for Riemann Hypothesis where $\sigma_1(n)=\sum_{d|n}d$ is sum of divisors ...
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2 votes
0 answers
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On consecutive superabundant numbers

Define $\sigma(n)=\sum_{d\mid n} d$. A number $n>1$ is said to be superabundant (SA) if it is an integer and $\frac{\sigma(n)}{n}>\frac{\sigma(s)}{s}$ for every positive integer $s<n$. Let $n$...
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2 votes
0 answers
286 views

On odd perfect numbers and a GCD - Part II

(Note: A detailed version of this question was posted in MSE last April 15, 2020. It has not received any responses there as of yet. I have therefore cross-posted it here, hoping that it is ...
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7 votes
0 answers
214 views

Restricted divisor summatory function

I have a problem that boils down to prove that the succession $\{a_n\}$ tends to infinity, where $$a_n:=1+\sum_{0\leq j<n}D_{2j+1}(n-j)$$ and $D_{m}(n)$ is the number of divisors $d>1$ of $n$ ...
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2 votes
2 answers
544 views

On the nearest-square function and the quantity $m^2 - p^k$ where $p^k m^2$ is an odd perfect number

This question has been cross-posted from this MSE question and is an offshoot of this other MSE question. (Note that MSE user mathlove has posted an answer in MSE, which I could not completely ...
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2 votes
0 answers
81 views

Does there exist a natural number $m$ such that $\sigma^{(k)}((2m+1)^2)$ is an odd square number for all $k\ge 0$?

This question, comes out of a question in MSE and I hope it is ok to ask it here: Does there exist a natural number $m$ such that $\sigma^{(k)}((2m+1)^2)$ is an odd square number for all $k\ge 0$? ...
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1 vote
0 answers
146 views

Generalized Thomas Ordowski conjecture at OEIS sequence A002326

OEIS is the online encyclopedia of integer sequences, Here is the link to the sequence $A002326$: https://oeis.org/A002326 For $n\geq 0$, the $n$th term in the sequence is defined as: $a(n)$ equals ...
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5 votes
1 answer
267 views

Is the factorization of $a_m-a_n$ affected by the fact that $\Sigma \frac{1}{a_k}<+\infty$?

I would like to ask the following. Let $(a_n)$ be a sequence of natural numbers such that $\sum_{k=1}^{\infty}\frac{1}{a_k}$ converges. Is it true that for infinitely many $m$, there is a $n<m$ ...
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2 votes
1 answer
570 views

$\frac{\sigma(n)}{n} < e \ln \ln (n)$ is true?

In Guy Robin, Grandes valeurs de la fonction somme des diviseurs et hypothèse de Riemann, J. Math. Pures Appl. 63 (1984), 187–213 (pdf) we find the following result: If the Riemann hypothesis is true ...
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7 votes
0 answers
153 views

Occurrence of binary words in divisibility patterns

Given an integer $n \geq 1$, let $d_n : \mathbb{N}_{\geq 1} \to \{0,1\}$ be the coloring of the positive integers defined by $d_n(x) = 1$ if $x \mathbin{|} n$ and $d_n(x) = 0$ otherwise. In other ...
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5 votes
2 answers
382 views

Is there a similar formula like Ramanunjan's Eisenstein series identity for $\sum_{k=1}^{n-1}k^2 \sigma(k)\sigma(n-k)$?

This question is related to the last question about van der Pol's identity for the sum of divisors. In Touchard (1953) it is mentioned that the sum of divisors $\sigma(n)$ satisfies the following ...
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8 votes
0 answers
595 views

Van der Pol's identity for the sum of divisors and a quartic polynomial equation for odd perfect numbers

In Touchard (1953) it is mentioned that the sum of divisors $\sigma(n)$ satisfies the following recurrence relation ($n>1$): $$n^2(n-1) = \frac{6}{\sigma(n)} \sum_{k=1}^{n-1}(3n^2-10k^2)\sigma(k)\...
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5 votes
0 answers
135 views

Touchard / van der Pol's identity for the sum of divisors and an elliptic curve for perfect numbers

In Touchard (1953) it is mentioned that the sum of divisors $\sigma(n)$, satisifies the following recurrence relation ($n>1$): $$n^2(n-1) = \frac{6}{\sigma(n)} \sum_{k=1}^{n-1}(3n^2-10k^2)\sigma(k)\...
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5 votes
0 answers
438 views

Does the equation $\sigma(\sigma(x^2))=2x\sigma(x)$ have any odd solutions?

This question was posted in MSE in early August 2020. It did garner several upvotes, but did not receive any responses. I have therefore cross-posted it here, hoping that it gets answered. Let $\...
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2 votes
1 answer
151 views

Sum of divisors and unitary divisors as the eigenvalue and the spectral norm of some addition matrix?

Let $n$ be a natural number and $D_n$ be the set of divisors. We can make this set to a ring by observing that each divisor $d$ has $$0 \le v_p(d) \le v_p(n)$$ Hence we can add two divisors $d,e$ by ...
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2 votes
1 answer
154 views

Is the asymptotic density of positive integers $n$ satisfying $\gcd(n, \sigma(n^2))=\gcd(n^2, \sigma(n^2))$ equal to zero?

(This post is an offshoot of this MSE question.) Let $\sigma(x)$ denote the sum of divisors of $x$. (https://oeis.org/A000203) QUESTION Is the asymptotic density of positive integers $n$ satisfying $...
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6 votes
1 answer
339 views

If $n = 18k+5$ is composite, there are at least 9 divisors of $\phi(n)$ which do not divide $n-1$

If $n$ is a composite of the form $18k+5$, there at least 9 divisors of $\phi(n)$ which do not divide $n-1$. Is this true in general or if not, what is the smallest counter example? The conjecture has ...
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1 vote
0 answers
98 views

If $n$ is a multiperfect number, then necessarily does one of its prime factors $p$ satisfy $p \parallel n$?

My question is as in the title: If $n$ is a multiperfect number, then necessarily does one of its prime factors $p$ satisfy $p \parallel n$? I quote from an answer by Varun Vejalla to a closely ...
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6 votes
0 answers
261 views

Is there a positive odd $n$ such that $\sigma(\sigma(n)) = \sigma(\sigma(n)-n)+\sigma(n)$?

Let $\sigma(n)$ denote the sum of the divisors of $n$. (https://oeis.org/A000203) It is relatively easy to find numbers $n$ such that $f(g(n)) = g(f(n))$ where $f(n) = \sigma(n)$ and $g(n) = \sigma(n) ...
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6 votes
0 answers
192 views

On nontotient Fibonacci numbers

This question is related to sequence of numbers $t$ such that $F_{6t}$ is a nontotient where $F_n$ represents the sequence of Fibonacci numbers for $n\geq 0$. The online encyclopedia Wikipedia has the ...
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