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.
250 questions
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Small Bézout coefficients along coprime sequences approaching ratio 1
Let $m_k$ and $n_k$ be two increasing sequences of positive integers with $(m_k,n_k)=1$ and
$\lim_{k\rightarrow\infty}m_k/n_k=1$.
Is it true that for sufficiently large $k$ there exist positive ...
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Are there primitive weird numbers that are multiple of a cube of an odd prime?
A positive integer $n$ is weird if it is abundant and not semiperfect, i.e., it cannot be expressed as a sum of distinct proper divisors of $n$.
A trivial consequence of the definition of weird number ...
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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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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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Proof of every highly abundant number greater than 3 is even
A natural number $n$ is defined as a highly abundant number (HAN) if and only if the sum of its divisors $\sigma(n)$ is strictly greater than the sum of the divisors of any natural number $m$ smaller ...
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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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