All Questions
Tagged with divisors-multiples nt.number-theory
220 questions
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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 ...
- 35
-4
votes
0
answers
144
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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
306
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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{...
- 25.4k
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1
answer
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If $N = qn^2$ is an odd perfect number with $\gcd(q,n)=1$, is it possible to have $q + 1 = \sigma(n)$?
The title says it all.
Question
If $N = qn^2$ is an odd perfect number with Euler prime $q$ and $\gcd(q,n)=1$, is it possible to have $q + 1 = \sigma(n)$?
Heuristic
From the Descartes spoof, with ...
CommunityBot
- 1
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
$$\gcd(\sigma(q^k),\sigma(n^2))=\frac{\bigg(\gcd(n,\sigma(n^2))\bigg)^2}{\gcd(n^2,\sigma(n^2))...
8
votes
2
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722
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Is there a similar formula like Ramanunjan's Eisenstein series identity for $\sum_{k=1}^{n-1}k^2 \sigma(k)\sigma(n-k)$?
This question is related to the last question about van der Pol's identity for the sum of divisors.
In Touchard (1953) it is mentioned that the sum of divisors $\sigma(n)$ satisfies the following ...
- 3,144
2
votes
1
answer
161
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Closed form expression for this zeta-like series involving GCD and LCM
I am looking for a closed form for this function $\Lambda:\mathbb{Q}^+\to\mathbb{R}^+$:
$$\Lambda(q) = \sum_{m,n\geq 1}\left(\frac{q\wedge\frac{m}{n}}{q\vee\frac{m}{n}}\right)^\alpha\left(\frac{m \...
- 8,374
12
votes
1
answer
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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:
$...
- 3,791
15
votes
3
answers
1k
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Do we know any bound on $\operatorname{lcm}(2^1-1, 2^2-1,\dots,2^n-1)$?
$\DeclareMathOperator\lcm{lcm}$We know that $\operatorname{lcm}(1,\dotsc,n)$ is approximately $e^n$ and we also know that $\gcd(2^a-1, 2^b-1)=2^{\gcd(a,b)}-1$.
I wonder if there exists an upper bound/...
- 34.4k
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 $\...
- 63.9k
3
votes
0
answers
121
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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 $n$...
- 101
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3
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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,...
- 149k
7
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1
answer
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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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2
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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 ...
- 5,409
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
$...
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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
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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 ...
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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 ...
3
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4
answers
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A conjecture regarding odd perfect numbers
(Note: I asked this question in MSE this June 2018 but did not receive any responses there. I have therefore cross-posted it here, hoping that it gets answered.)
Let $\sigma(z)$ denote the sum of ...
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 ...
- 18.7k
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 ...
- 105k
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votes
1
answer
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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 ...
- 4,713
7
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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 ...
- 433
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votes
2
answers
2k
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Is every prime the largest prime factor in some prime gap?
Definition: In the gap between any two consecutive odd primes we have one or more composite numbers. One of these composite number will have a prime factor which is greater than that of any other ...
- 114k
1
vote
0
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103
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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
$$
R(n,q,z)=(q+1)R(n-1,q+1,z)+\sum\limits_{j=0}^{q}...
- 4,948
2
votes
0
answers
158
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Exponential sum of $k$-fold divisor function
Can anyone point me to a reference for the main term when approximating the exponential sum of the 3-fold divisor function? Specifically I want the main term in $$\sum _{n\leq x}d_3(n)e\left (an/q\...
- 1,381
6
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1
answer
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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 ...
- 7,893
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7
answers
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Getting a bound on the coefficients of the factor polynomial
Suppose $f(x):=a_0+a_1x+\cdots+a_nx^n$ is a polynomial in $\mathbb{Z}[x]$ and $|a_i|\leq M$ for each $i=0,\ldots ,n.$ Now suppose $g(x)$ is a factor of $f(x)$ in $\mathbb{Z}[x]$, then is it possible ...
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2
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2
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484
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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 ...
1
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0
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167
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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, ...
0
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1
answer
203
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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) - ...
1
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1
answer
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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
$$\gcd(\sigma(q^k),\sigma(n^2)) = \frac{(\gcd(n,\sigma(n^...
3
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1
answer
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On the OEIS sequence A327265
The OEIS sequence https://oeis.org/A327265 starts:
$$1, 2, 5, 11, 19, 31, 51, 89, 123, 151, 179, 181, 180, 365, 634, 657, 656, 655.$$
$\mathrm{A327265}(n)$ is the smallest $k$ such that $\mathrm{...
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1
answer
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Is the divisor counting function equidistributed mod $p$?
Let $\sigma_0(n)$ be the divisor counting function:
$$\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 ...
- 12.9k
4
votes
4
answers
461
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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) ...
CommunityBot
- 1
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3
answers
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On Robin's criterion for RH [closed]
\begin{equation}
\sigma(n) < e^\gamma n \log \log n
\end{equation}
In 1984 Guy Robin proved that the inequality is true for all n ≥ 5,041 if and only if the Riemann hypothesis is true (Robin 1984)....
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0
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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 ...
0
votes
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
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76
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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)]$ ...
- 433
3
votes
2
answers
795
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Estimate about primes
Can anyone give an estimate (upper bound or lower bound) for the number of divisors $d\mid P_r$ such that $\frac{\sqrt{P_r}}{2}< d < \sqrt{P_r}$, where $P_r$ is the product of the $r$ smallest ...
- 4,713
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
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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}.$$
(Note that it is known that $v \geq 9$ by work of Nielsen.) Let $N(m)$ be the number of ...
1
vote
1
answer
153
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Number of distinct near-squares primes dividing an odd perfect number
I'm curious about if the following question is in the literature or what work can be done about it.
Denote the number of distinct primes dividing an odd perfect number $N$ with the arithmetic function ...
- 6,969
4
votes
1
answer
263
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Divisibility relation with a specific sum of divisors
Let $\sigma(n)$ be the sum of the divisors of $n$. Is it always true that if $n$ is odd, that $$n\mid\sum_{k=1}^{\frac{n-1}{2}}k^2\sigma(k)\sigma(n-k)?$$
I have checked this up to $n=100$, and I ...
- 1,755
1
vote
0
answers
69
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Shapiro inequality for divisor sets
The Shapiro inequality is the statement that if $x_1, x_2, \dots, x_n$ are positive, with $x_{n+1}=x_1, x_{n+2}=x_2$, then $$\sum_{i=1}^{n} \frac{x_i}{x_{i+1}+x_{i+2}} \geq \frac{n}{2}.$$ This can be ...
- 63.9k
8
votes
1
answer
297
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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 ...
- 14.7k
-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 ...
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4
votes
1
answer
129
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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 ...
- 109k
2
votes
1
answer
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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 ...
2
votes
0
answers
342
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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 ...