All Questions
Tagged with divisors-multiples nt.number-theory
220 questions
-4
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0
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Divisors of n and n + 1
Suppose $a$ is a proper divisor of $n$ (where $n$ is a positive integer), and $b$ a proper divisor of $n + 1$.
Is there a general criterion (or general property of $n$) which enables one to conclude ...
- 4,605
8
votes
1
answer
305
views
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{...
- 25.4k
2
votes
1
answer
161
views
Closed form expression for this zeta-like series involving GCD and LCM
I am looking for a closed form for this function $\Lambda:\mathbb{Q}^+\to\mathbb{R}^+$:
$$\Lambda(q) = \sum_{m,n\geq 1}\left(\frac{q\wedge\frac{m}{n}}{q\vee\frac{m}{n}}\right)^\alpha\left(\frac{m \...
- 634
16
votes
4
answers
2k
views
+100
Square roots and prime numbers
Definitions:
Here I present a novel conjecture using basic mathematical tools like the sum of the
divisors of an integer $n$ called $\sigma(n)$, the sum of the squares of the positive divisors of n ...
- 35
1
vote
1
answer
78
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Minimum value of a function involving the divisor counting function
Fix any positive integer $n\in\mathbb{Z}^+,$ and consider the function $f_n : \mathbb{Z}^+\setminus\{n\}\to\mathbb{Z}^+$ given by $$f_n(t)=\sigma_0(n)+\sigma_0(t)-2\sigma_0(\gcd(n, t)),$$ where $\...
- 1,093
12
votes
3
answers
715
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When does $2$ arise when using the Euclidean algorithm to compute greatest common divisors?
When using the standard Euclidean algorithm to compute the greatest common divisor of a pair of relatively prime positive integers, the integer $2$ sometimes arises and sometimes does not. For example,...
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1
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0
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61
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Is $N - \varphi(N)$ a square, if $N = q^k m^2$ is an odd perfect number with special prime $q$?
This question was inspired by this MSE question.
In MSE, it is shown that
$$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 $...
2
votes
1
answer
202
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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
$...
- 253
-4
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2
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173
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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 ...
2
votes
1
answer
137
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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 ...
0
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0
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55
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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 ...
12
votes
1
answer
2k
views
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:
$...
- 45
6
votes
2
answers
685
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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},$$
the number of divisors of $x$ which are at most $n$. Question 6 of ...
- 541
0
votes
1
answer
418
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On a GCD approach to odd perfect numbers
Let $N = p^k m^2$ be an odd perfect number with special prime $p$ satisfying $p \equiv k \equiv 1 \pmod 4$ and $\gcd(p,m)=1$. Let $\sigma(z)$ denote the classical sum of divisors of the positive ...
1
vote
0
answers
103
views
Mysterious recursion for the A005225
Let $a(n)$ be A005225 i.e. number of permutations of length $n$ with equal cycles. Here
$$
a(n)=n!\sum\limits_{d|n}\frac{1}{d!(\frac{n}{d})^d}
$$
Let
$$
R(n,q,z)=(q+1)R(n-1,q+1,z)+\sum\limits_{j=0}^{q}...
- 4,948
8
votes
1
answer
205
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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 ...
- 125
2
votes
0
answers
158
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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 $$\sum _{n\leq x}d_3(n)e\left (an/q\...
- 1,381
7
votes
2
answers
1k
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Convolution sum of divisor functions
Let $\sigma_0(n)$ be the divisor counting function
$$\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)$$
I ran some quick ...
- 1,949
9
votes
1
answer
558
views
Is the divisor counting function equidistributed mod $p$?
Let $\sigma_0(n)$ be the divisor counting function:
$$\sigma_0(n) = \sum_{d \vert n} 1.$$
I ran some numerical experiments that showed when $p$ is prime, the function $\sigma_0(n)$ is equidistributed ...
- 1,949
1
vote
0
answers
167
views
On "Euclidean" odd perfect numbers
In what follows, we let $N = r^s u^2$ be an odd perfect number given in Eulerian form, i.e. $r$ is the special prime satisfying $r \equiv s \equiv 1 \pmod 4$ and $\gcd(r,u)=1$. In this preprint, ...
4
votes
0
answers
87
views
On Carmichael function and aliquot parts of odd perfect numbers
I've asked nine months ago this question on Mathematics Stack Exchange with identifier 4430381 and same title. There is not answer for this question on Mathematics Stack Exchange, I wondered if this ...
3
votes
1
answer
1k
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On the OEIS sequence A327265
The OEIS sequence https://oeis.org/A327265 starts:
$$1, 2, 5, 11, 19, 31, 51, 89, 123, 151, 179, 181, 180, 365, 634, 657, 656, 655.$$
$\mathrm{A327265}(n)$ is the smallest $k$ such that $\mathrm{...
user178594
0
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1
answer
114
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Given that $H = \frac{n^2}{\sigma(q^k)/2} = G \times J^2$, where $q^k n^2$ is an odd perfect number, then what is the value of $\gcd(G, J)$?
Let $N = q^k n^2$ be an odd perfect number with special prime $q$ satisfying $q \equiv k \equiv 1 \pmod 4$ and $\gcd(q,n)=1$. Denote the classical sum of divisors of the positive integer $x$ by $\...
3
votes
0
answers
76
views
Divisor of given order in short intervals
Is the following Open question or Conjecture already known, or eventually settled ?
Open question : For sufficiently large $x$ there is at least a positive integer in the interval $[x,x+\log^2(x)]$ ...
- 433
4
votes
0
answers
117
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Greatest common divisors of some binomial coefficients
This is cross-posted from math.stackexchange.
While making some computation, I stumbled upon a curious relation among some binomial coefficients.
Consider the sequence of binomial coefficients $a(k,n)$...
- 700
4
votes
1
answer
263
views
Divisibility relation with a specific sum of divisors
Let $\sigma(n)$ be the sum of the divisors of $n$. Is it always true that if $n$ is odd, that $$n\mid\sum_{k=1}^{\frac{n-1}{2}}k^2\sigma(k)\sigma(n-k)?$$
I have checked this up to $n=100$, and I ...
- 6,969
1
vote
0
answers
69
views
Shapiro inequality for divisor sets
The Shapiro inequality is the statement that if $x_1, x_2, \dots, x_n$ are positive, with $x_{n+1}=x_1, x_{n+2}=x_2$, then $$\sum_{i=1}^{n} \frac{x_i}{x_{i+1}+x_{i+2}} \geq \frac{n}{2}.$$ This can be ...
- 6,969
8
votes
1
answer
297
views
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 ...
- 3,499
4
votes
4
answers
461
views
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) ...
4
votes
1
answer
129
views
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 ...
-10
votes
1
answer
556
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Arithmetic billiards, prime numbers and the Goldbach conjecture
I've edited the following post on Mathematics Stack Exchange, (now closed, at that date I'm suspended) with identifier 4510963, please let me to know if you've some doubt or I can improve the post.
On ...
1
vote
1
answer
153
views
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 ...
0
votes
1
answer
203
views
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) - ...
3
votes
1
answer
213
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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$ ...
5
votes
1
answer
187
views
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)$?
- 613
2
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1
answer
380
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A conjecture concerning the equation $\sigma\left(\square\right)=\text{prime}$
I can deduce the following simple proposition, the definitions for $\sigma(x)$ the sum of divisors functions and $\varphi(x)$ the Euler totient function are assumed. After I present a conjecture that ...
1
vote
0
answers
153
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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$ ...
4
votes
1
answer
271
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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 ...
2
votes
1
answer
307
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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}) ...
- 5,470
2
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0
answers
108
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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$...
- 523
3
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0
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180
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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 ...
3
votes
2
answers
234
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 ...
- 133
3
votes
0
answers
179
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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 ...
- 1,381
3
votes
0
answers
82
views
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
$$
S_a(x):=\sum_{n \in A} d(n)^a \geq |A|(\ln x)^{...
- 10.4k
5
votes
1
answer
264
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Analogue of the second Hardy-Littlewood conjecture for numbers of divisors?
Let $f(n)$ denote the proposition "There exists some $k>1$ such that
$$
\sum_{m=k}^{k+n-1}\tau(m) < \sum_{m=1}^n\tau(m)
$$
where $\tau(m)$ is the number of the divisors of $m$." (This ...
- 9,114
1
vote
0
answers
305
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About inequalities that involve the sum of divisors, the Euler's totient and the aliquot part $\sigma(n)-n$
In this post, for integers $n\geq 1$, I denote the sum of divisors $\sum_{1\leq d\mid n}d$ as $\sigma(n)$ and the Euler's totient function as $\varphi(n)$. It's easy to check* that if we assume that $...
2
votes
2
answers
484
views
On odd perfect numbers $p^k m^2$ with special prime $p$ satisfying $m^2 - p^k = 2^r t$ - Part II
(Preamble: We have asked this same question in MSE two weeks ago, without getting any answers. We have therefore cross-posted it to MO, hoping that it gets answered here.)
The topic of odd perfect ...
0
votes
0
answers
107
views
On improving the upper bound $I(m^2) \leq \frac{2p}{p+1}$, if $p^k m^2$ is an odd perfect number with special prime $p$
(Preamble: This question is an offshoot of this answer to an MSE question with the same title.)
Denote the classical sum of the divisors of the positive integer $x$ by $\sigma(x)=\sigma_1(x)$ and the ...
2
votes
0
answers
751
views
Does $n \mid \sigma(n^2)$, if $q^k n^2$ is an odd perfect number?
Let $\sigma(x)=\sigma_1(x)$ be the classical sum of divisors of the positive integer $x$.
It is known that
$$\gcd(\sigma(q^k),\sigma(n^2))=\frac{\bigg(\gcd(n,\sigma(n^2))\bigg)^2}{\gcd(n^2,\sigma(n^2))...
1
vote
1
answer
321
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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^...