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.
248 questions
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On the non-squarefree odd part of a weird number
A positive integer $n$ is weird if it is abundant and cannot be expressed as a sum of distinct proper divisors of $n$. As in the case of perfect numbers, all weird numbers currently known are even (in ...
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$O(1)$ algorithm for factoring integers of the form $n=X (X^D+O(X^{D-1}))$
Factorization of integers of special forms are of both theoretical
interest and cryptographic implications.
Experimentally we found a seemingly "large" set of integers for which a divisor ...
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Is the least common multiple sequence $\text{lcm}(1, 2, \dots, n)$ a subset of the highly abundant numbers?
I've been comparing the sequence of the Least Common Multiple of the first $n$ integers, $L_n = \text{lcm}(1, 2, \dots, n)$, with the sequence of Highly Abundant Numbers (HA).
The two sequences in ...
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Divisibility property of colossally abundant numbers
The sequence of colossally abundant (CA) numbers, $a(n)$ (OEIS A004490), consists of positive integers that maximize the ratio $\frac{\sigma(m)}{m^{1+\epsilon}}$ for some $\epsilon > 0$.
A known ...
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Existence of an Integer Sequence with Specific Properties Related to Divisors and Fractional Parts
Motivated by a problem concerning the existence of quasi-periodic solutions in dynamical systems, I encountered a number-theoretic question that appears somewhat unusual. I would like to know whether ...
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Upper bounds on the greatest common divisor of sums of geometric series
(cross-posted from Math Stack Exchange, as it did not get any comment or answer, even after being bountied)
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^{...
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Is it possible for a polynomial of the form $P(x^b)$ to share a root with $F(x^b) + R(x)$?
Let $P(x)$ be a polynomial in one variable with coefficients in a number field $K$. Let $b \geq 2$ be a positive integer. Suppose $P(x^b)$ is irreducible. Let $F(x)$ and $R(x)$ be polynomials with ...
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Applications of Bang's lemma as in the solution to Tarski's problem to a number theoretic problem?
Inspired by this question and answer, I want to apply Bang's lemma to a number theoretic setting:
Bang's lemma for p.d. kernels: Let $k : X \times X \rightarrow \mathbb{R}$ be a positive definite ...
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Gap in binomial coefficients
Let $a,b$ be positive integers. Because binomial coefficients are integers, we know that $a!b!$ divides $(a+b)!$. For particular $a$ and $b$ there may be a gap $g$ with a tighter result, so $a!b!$ ...
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Bounded gaps yield chains of which length w.r.t. divisibility
A well-known application of the pigeonhole principle, due to Erdős, says that every subset of $\{1,2,\dots, 2n\}$ of size $n+1$ contains two distinct elements dividing each other. This remark can be ...
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Furstenberg's topological proof method (of prime infinitude) might be more powerful than initially thought. $\sigma_0(\frac{p_n - 1}{4})$ odd i.o.?
Let $\sigma_0(n)$ be from number theory, i.e. the total number of divisors of an integer $n$. Let $p_n$ denote the $n$th prime number.
Let $S =$ the set of prime numbers $p \in 4\Bbb{Z} + 1$ such ...
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Is there a Carmichael number $n$ satisfying $\sigma(n)\equiv0 \pmod{n+1}$?
Question: Is there a Carmichael number $n$ satisfying $\sigma(n)\equiv0\pmod{n+1}$ ?
Motivation: It can be proved that a positive integer $n$ simultaneously satisfying $n\equiv1\pmod{\phi(n)}$ and $\...
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On bounds for the factors of integers
Let $N=p q$ and let $D=\frac{\log{q}}{\log{p}} > 1$.
Conjecture 1 There exist positive real constant $A$ depending
only on $D$ such that given integer $r$ in the range
$[q-q^{\frac{D-1}{D}},q+q^{\...
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Identity?: $\frac{\varphi(2^n-1)}{n}=\frac{2^{\varphi(2^n-1)}-1 \bmod (2^n-1)^2}{2^n-1}$
The computer found this.
Let $n$ be a positive integer.
Up to $n=200$ we have:
$$\frac{\varphi(2^n-1)}{n}=\frac{2^{\varphi(2^n-1)}-1 \bmod (2^n-1)^2}{2^n-1}. \tag{1}\label{483144_1}$$
Q1 Is \eqref{...
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Closed form expression for this zeta-like series involving GCD and LCM
I am looking for a closed form for this function $\Lambda:\mathbb{Q}^+\to\mathbb{R}^+$:
$$\Lambda(q) = \sum_{m,n\geq 1}\left(\frac{q\wedge\frac{m}{n}}{q\vee\frac{m}{n}}\right)^\alpha\left(\frac{m \...
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Square roots and prime numbers
Definitions:
Here I present a novel conjecture using basic mathematical tools like the sum of the divisors of an integer $n$ called $\sigma(n)$, the sum of the squares of the positive divisors of $n$ ...
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Minimum value of a function involving the divisor counting function
Fix any positive integer $n\in\mathbb{Z}^+,$ and consider the function $f_n : \mathbb{Z}^+\setminus\{n\}\to\mathbb{Z}^+$ given by $$f_n(t)=\sigma_0(n)+\sigma_0(t)-2\sigma_0(\gcd(n, t)),$$ where $\...
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When does $2$ arise when using the Euclidean algorithm to compute greatest common divisors?
When using the standard Euclidean algorithm to compute the greatest common divisor of a pair of relatively prime positive integers, the integer $2$ sometimes arises and sometimes does not. For example,...
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Is $N - \varphi(N)$ a square, if $N = q^k m^2$ is an odd perfect number with special prime $q$?
This question was inspired by this MSE question.
In MSE, it is shown that
$$n - \varphi(n) = (2^{p-1})^2$$
if $n = {2^{p-1}}(2^p - 1)$ is an even perfect number.
Here is my question in this post:
Is $...
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Exponential sums involving smooth truncated divisor functions
Let $p$ be a prime, $a \neq 0$ an integer, let $M,N \gg 1$ and let $\psi,\eta$ be some fixed Schwartz functions. Would you know of any references in the literature where upper bounds for sums such as
$...
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If $p^k m^2$ is an odd perfect number with special prime $p$, then $p^k < 2am$ for some positive integer $a < m$ [closed]
(Preamble: Andy Putman asserts, in the comments, that MO policy prohibits "requests to check completeness of proofs". I have therefore trimmed down my original question to the bare ...
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If $p^k m^2$ is an odd perfect number with special prime $p$, then must $m^2 - p^k = s^2 - t^2$ hold for some $s$ and $t$?
My present question is as is in the title:
If $p^k m^2$ is an odd perfect number with special prime $p$, then must $m^2 - p^k = s^2 - t^2$ hold for some $s$ and $t$?
It is known that $m^2 - p^k$ is ...
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If $p^k m^2$ is an odd perfect number with special prime $p$, is it possible to have $p = k$?
Denote the classical sum of divisors of the positive integer $x$ by $\sigma(x)=\sigma_1(x)$.
My question is as is in the title:
If $p^k m^2$ is an odd perfect number with special prime $p$, is it ...
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Power of primes
$n$ is a natural number $>1$, $\varphi(n)$ denotes the Euler's totient function, $P_n$ is the $n^\text{th}$ prime number and $\sigma(n)$ is the sum of the divisors of $n$. Consider the expression:
$...
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Number of divisors which are at most $n$
I’m interested in the function $\tau_n:\mathbb{N}\to\{1,2,3,\cdots, n\}$ defined by
$$\tau_n(x)=\sum_{k=1}^n \mathbf{1}_{k\mid x},$$
the number of divisors of $x$ which are at most $n$. Question 6 of ...
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On a GCD approach to odd perfect numbers
Let $N = p^k m^2$ be an odd perfect number with special prime $p$ satisfying $p \equiv k \equiv 1 \pmod 4$ and $\gcd(p,m)=1$. Let $\sigma(z)$ denote the classical sum of divisors of the positive ...
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What can be said about $\gcd(N/q^{\alpha},\sigma(N/q^{\alpha}))$ where $N$ is an odd perfect number and $q^{\alpha} \parallel N$?
What can be said about the quantity $$\gcd(N/q^{\alpha},\sigma(N/q^{\alpha}))$$ where $N$ is an odd perfect number and $q^{\alpha} \parallel N$? In particular, can one prove that it is always greater ...
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Representation of a number as a product of $\sqrt{n^2 + 1} + n$
Question. Do there exist two multisets $A, B$ consisting of positive integer numbers such that $|A|$ and $|B|$ have different parity and
$$
\prod_{n\in A}(n + \sqrt{n^2 + 1}) = \prod_{m\in B}(m + \...
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Are there infinite numbers of the form $\sigma_1(n)=\sigma_1(m)=p$, or is there only one?
I put forward a hypothesis in number theory, it is as follows.$ \sigma_1(n)=\sigma_1(m)=p$, where $\sigma_1$ is the divisor sum function, $n,m\in \mathbb N$, and $p$ is prime. I recently noticed and ...
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Exponential sum of $k$-fold divisor function
Can anyone point me to a reference for the main term when approximating the exponential sum of the 3-fold divisor function? Specifically I want the main term in $$\sum _{n\leq x}d_3(n)e\left (an/q\...
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Bounds of heights of coefficients of rational polynomials
For a non zero rational $r=p/q$ ($p,q\in\mathbb Z$ coprimes), define the height of $r$ by $\mathrm{ht}(r)=\max(|p|,|q|)$ (by convention $\mathrm{ht}(0)=0$). For a polynomial $P\in\mathbb Q[X]$, define ...
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Convolution sum of divisor functions
Let $\sigma_0(n)$ be the divisor counting function
$$\sigma_0(n) = \sum_{d \vert n} 1.$$
I'm interested in the convolution sum
$$ S(n) := \sum_{k=1}^{n-1} \sigma_0(k) \sigma_0(n-k)$$
I ran some quick ...
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Is the divisor counting function equidistributed mod $p$?
Let $\sigma_0(n)$ be the divisor counting function:
$$\sigma_0(n) = \sum_{d \vert n} 1.$$
I ran some numerical experiments that showed when $p$ is prime, the function $\sigma_0(n)$ is equidistributed ...
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On "Euclidean" odd perfect numbers
In what follows, we let $N = r^s u^2$ be an odd perfect number given in Eulerian form, i.e. $r$ is the special prime satisfying $r \equiv s \equiv 1 \pmod 4$ and $\gcd(r,u)=1$. In this preprint, ...
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On Carmichael function and aliquot parts of odd perfect numbers
I've asked nine months ago this question on Mathematics Stack Exchange with identifier 4430381 and same title. There is not answer for this question on Mathematics Stack Exchange, I wondered if this ...
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On the OEIS sequence A327265
The OEIS sequence https://oeis.org/A327265 starts:
$$1, 2, 5, 11, 19, 31, 51, 89, 123, 151, 179, 181, 180, 365, 634, 657, 656, 655.$$
$\mathrm{A327265}(n)$ is the smallest $k$ such that $\mathrm{...
user178594
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Given that $H = \frac{n^2}{\sigma(q^k)/2} = G \times J^2$, where $q^k n^2$ is an odd perfect number, then what is the value of $\gcd(G, J)$?
Let $N = q^k n^2$ be an odd perfect number with special prime $q$ satisfying $q \equiv k \equiv 1 \pmod 4$ and $\gcd(q,n)=1$. Denote the classical sum of divisors of the positive integer $x$ by $\...
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Divisor of given order in short intervals
Is the following Open question or Conjecture already known, or eventually settled ?
Open question : For sufficiently large $x$ there is at least a positive integer in the interval $[x,x+\log^2(x)]$ ...
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Divisibility of Stirling numbers
It is well known that if $p$ is prime, Stirling numbers of the first and second kind, $s_1(p,k)$ and $s_2(p,k)$, are divisible by $p$ if $1<k\le p-1$ (Lagrange ; easiest is working in $\mathbb F_p$ ...
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Greatest common divisors of some binomial coefficients
This is cross-posted from math.stackexchange.
While making some computation, I stumbled upon a curious relation among some binomial coefficients.
Consider the sequence of binomial coefficients $a(k,n)$...
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When an element of a ring that is divisible by a finite set of elements is necessarily divisible by their product?
In a commutative ring $R$, when does the assumption $r_i\mid r$ for $1\le i\le n$ imply $\prod_{1\le i\le n} r_i\mid r$ (when $r_i$ are fixed)?
Does there exist any criterion for this implication that ...
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Divisibility relation with a specific sum of divisors
Let $\sigma(n)$ be the sum of the divisors of $n$. Is it always true that if $n$ is odd, that $$n\mid\sum_{k=1}^{\frac{n-1}{2}}k^2\sigma(k)\sigma(n-k)?$$
I have checked this up to $n=100$, and I ...
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Shapiro inequality for divisor sets
The Shapiro inequality is the statement that if $x_1, x_2, \dots, x_n$ are positive, with $x_{n+1}=x_1, x_{n+2}=x_2$, then $$\sum_{i=1}^{n} \frac{x_i}{x_{i+1}+x_{i+2}} \geq \frac{n}{2}.$$ This can be ...
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Divisibility chains and polynomials
Let $P\in \Bbb{Z}[X]$ be a polynomial with degree $d>1$.
It is conjectured that for all such $P$, their range for integer inputs $R_P:=P(\Bbb{Z})$ has finite intersection with the set of factorials ...
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On the diophantine equation $x^{m-1}(x+1)=y^{n-1}(y+1)$ with $x>y$, over integers greater or equal than two
I've asked two years ago a post on Mathematics Stack Exchange, were provided two excellent answers. I'm asking on MathOverflow in the hope that some professor can to expand/improve (if it is possible) ...
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Maximal number of divisors of numbers whose sum does not exceed $n$
Denote by $f(n)$ the maximal number of distinct divisors of $k$ integer numbers $1\leq a_1<a_2<\ldots<a_k\leq n$, where $k$ is not fixed and $a_1+\ldots+a_k\leq n$. I'm interested in the ...
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Number of distinct near-squares primes dividing an odd perfect number
I'm curious about if the following question is in the literature or what work can be done about it.
Denote the number of distinct primes dividing an odd perfect number $N$ with the arithmetic function ...
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If $p^k m^2$ is an odd perfect number with special prime $p$, then under what other conditions on $\sigma(p^k)/2$ does $k=1$ follow?
Let $N = p^k m^2$ be an odd perfect number with special prime $p$ satisfying $p \equiv k \equiv 1 \pmod 4$ and $\gcd(p,m)=1$.
Descartes (1638), Frenicle (1657), and subsequently [Sorli (2003) - ...
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A diophantine equation inspired in a conjecture due to Gica and Luca, example of a large Mersenne exponent
In this post I consider the equation $$k\cdot x=y^2+z^2(x^2-2)-2\tag{1}$$
over odd integers $y\geq 1$ and $z\geq 1$, and over integers $k\geq 1$ and very large Mersenne exponents $x$ such that $x^2-2$ ...
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Small covering of divisors
Let $D_n$ be the set of divisors of $n$.
Does there always exists a $B\subseteq D_n$ such that $D_n = \{\gcd(ab,n) \mid a\leq \sqrt{n}, b\in B\}$ and $\sum_{b\in B} \frac{n}{b}=O(n)$?
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