**-1**

votes

**0**answers

71 views

### Sum-free sets of powerful numbers

For $n=p_1^{\alpha_1}\cdots p_r^{\alpha_r}$ with distinct primes $p_i$, call $\alpha= (\alpha_1,\dots,\alpha_r)$ the type of $n$ and denote by $N_\alpha$ the set of all naturals of this type.
We ...

**2**

votes

**1**answer

89 views

### On attempting a proof for $r > 1$, if $M = {2^r}{b^2}$ is an even almost perfect number which is not a power of two

(Preamble: I first thought that this question might be more appropriate for MSE. However, I posted it here nonetheless in the hope that someone with that brilliant idea can help with answering my ...

**1**

vote

**1**answer

120 views

### Is it possible to have an even superperfect number and an odd superperfect number whose product is an almost perfect number?

A number $n \in \mathbb{N}$ is said to be superperfect if
$$\sigma(\sigma(n)) = 2n.$$
A number $m \in \mathbb{N}$ is said to be almost perfect if $$\sigma(m) = 2m - 1.$$
Here is my question:
Is ...

**1**

vote

**0**answers

176 views

### On even almost perfect numbers other than powers of two

(Note: This question is an improved version of and has been cross-posted from this MSE post.)
Let $\sigma(x)$ denote the sum of the divisors of $x$. If $\sigma(x) = 2x - 1$, then we call $x$ an ...

**8**

votes

**0**answers

250 views

### Has it been proved that odd perfect numbers cannot be triangular?

(Note: This question has been cross-posted from MSE.)
Euclid and Euler proved that every even perfect number is of the form $m = \frac{{M_p}\left(M_p + 1\right)}{2}$ where $M_p = 2^p - 1$ is a prime ...

**0**

votes

**1**answer

102 views

### What is the relative size of the radical of an ABC-triple relative to the number of primes up to its largest element?

Write $\bf N$ for the set of natural numbers, and $P$ for the set of primes. For $x$ in $\bf N$ let $p(x)$ be the product of the primes dividing $x$ (that is, the "radical" of $x$). Also write $\#(x)$ ...

**0**

votes

**0**answers

44 views

### If $N = {q^k}{n^2}$ is an odd perfect number given in Eulerian form, is $n^2$ solitary?

(Note: A similar question was asked in MSE two months ago.)
Let $\sigma(x)$ be the sum of the divisors of the natural number $x$, and denote the abundancy index $\sigma(x)/x$ by $I(x)$.
Here is my ...

**1**

vote

**1**answer

402 views

### If $N = qn^2$ is an odd perfect number, 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 $q$, is it possible to have $q + 1 = \sigma(n)$?
Heuristic
From the Descartes spoof, with quasi-Euler ...

**0**

votes

**2**answers

171 views

### Is there an example of integers ($x,p, q ,y$ ) which satisfies the below conditions in this claim? [closed]

Edit 01:In order to look divisibility among power divisor function where i would like to know if there a such integer $n>1 $ with y coprime to $x$ then we have: :$\sigma_y(n)\bmod \sigma_x(n)=0$, ...

**0**

votes

**1**answer

184 views

### For which $x$ and $y$ does $\sigma_x(n) $ divide $\sigma_y(n)$ for all $n$?

I would like to know more about divisibility among power-divisor functions. Put $\sigma_k(n) = \sum_{d \mid n} d^k$ for all positive integers $k$ and $n$.
My question here is : for which positive ...

**0**

votes

**0**answers

96 views

### How do divisors of smooth numbers clump together?

Simply put, let $n$ have $p=n^r$ be its largest prime factor, with $r\lt 1/j$ for some integer $j$. For real $t \in [0,1-r]$, let $f(t)$ be the fraction of divisors $d$ of $n$ with $d \in [n^t,n^{t+r}...

**4**

votes

**1**answer

177 views

### Improvement of a bound on divisor distributions from “Divisors” (Hall and Tenenbaum)?

In the classic text referred to in the title of this question, the bound
$$
H(x,y,2y) \ll \frac{x}{(\log y)^{\delta}\sqrt{\log \log y}},\quad (3\leq y\leq \sqrt{x})
$$
is given, where $\delta=1-\frac{...

**0**

votes

**0**answers

84 views

### Sum of reciprocals of primitive sequences with distinct prime factors

In a previous mathoverflow question here a construction of a primitive sequence $1<a_1<\cdots<a_k\leq n$ formed by including all the integers in $[1,n]$ with exactly $k$ prime divisors (...

**1**

vote

**1**answer

239 views

### Some divisibility constraints in Frobenius coin problem

Let's say that the linear form $ax+by$ represents $n$ if $ax+by=n$ for some positive integer $x$ and $y$.
Call a pair $(a,b)\in\Bbb N\times\Bbb N$ with $\mathsf{gcd}(a,b)=1$ excellent if linear form ...

**3**

votes

**1**answer

178 views

### Primitive sequence $a_i$ attaining Pillai's bound on $\sum_{i} 1/a_i$

A primitive sequence $1<a_1<\ldots<a_k\leq n$ is a sequence of integers no one of which divides any other, investigated by Erdos, Behrend and others, over the last 80 years. In fact, $\max k=\...

**3**

votes

**1**answer

260 views

### Problem related to Frobenius coin problem

Let's say that the linear form $ax+by$ represents $n$ if $ax+by=n$ for some positive integer $x$ and $y$.
Call a pair $(a,b)\in\Bbb N\times\Bbb N$ with $\mathsf{gcd}(a,b)=1$ good if,
for any $r,s,u,...

**1**

vote

**0**answers

80 views

### Consecutive integers divisible by consecutive small numbers

Given $n$, what is the largest set of consecutive integers in $[n,2n]$ can we have so that each integer is divisible by a distinct element from $[\log n,2\log n]$ (no partiular order)? So apriori I am ...

**0**

votes

**0**answers

55 views

### $\mathsf{GCD}$s of random linear form

Given $a,b\in\Bbb N_{<M}$ where $M\in\Bbb N_{>\exp(18)}$ is arbitrary with $(a,b)=1$, the probability that $\mathsf{gcd}(ax_1+by_1,ax_2+by_2)=1$ where $x_1,x_2,y_1,y_2\in\Bbb N_{>\ln M}$ is ...

**13**

votes

**2**answers

515 views

### Number of distinct factors

Denote $\omega(m)$ to be number of distinct factors of $m$ as defined in http://mathworld.wolfram.com/DistinctPrimeFactors.html.
At every $c>0$, given $n\in\Bbb N$ define $$S(n,c)=\big\{m\in\Bbb N:...

**0**

votes

**1**answer

259 views

### Are there infinitely many $k$ for which $\frac{\sigma(k)}{k}=n^p$ and $p$ is an odd prime? [closed]

I would like to know if there are infinitely many $k$ for which $$\sigma(k)/k=n^p$$ such that $m=k{n}^{p-1}$ with $m,n>0$ and $p$ is an odd prime?
Note: $\sigma(\frac{m}{{n}^{p-1}})$ is the sum of ...

**6**

votes

**1**answer

251 views

### Is $\liminf \frac{\sigma_{k}(n)}{n}$ finite for every $k$?

Can someone show me how to prove that $$\liminf_{n \to \infty} \frac{\sigma_{k}(n)}{n} < \infty$$ for every natural number $k$? Or is this problem open?
Here, $\sigma_{k}(n)=\sigma(\sigma(\sigma(\...

**8**

votes

**2**answers

1k views

### Is there a fixed integer $x>1$ satisfing ${\sigma}^{k}(x)\equiv 0\pmod{x}$ for all positive integers $k$?

This question related to this question from SE. I'm interested to know if there exists an integer $x>1$ that satisfies $${\sigma}^{k}(x)\equiv 0\pmod{x}$$
for all positive integers $k$.
Note. $\...

**10**

votes

**3**answers

542 views

### On the number of consecutive divisors of an integer

Define for $n \in \mathbb{N}$ the function $$\tau_1(n):=\sum_{\substack{d|n, \\ d+1|n}}1,$$ i.e. the number of consecutive divisors of an integer. The average of $\tau_1(n)$ is $1$ since $$\sum_{n\leq ...

**-2**

votes

**1**answer

199 views

### On odd perfect numbers $N$ given in the Eulerian form $N = {q^k}{n^2}$, Part II

I posted this question on MSE two days ago, but did not receive any responses. I have cross-posted it on MO, hoping it gets more attention here and that it is appropriate for this site.
A positive ...

**1**

vote

**1**answer

183 views

### Does there exist an integer that is both solitary and almost perfect?

This question is an offshoot from the following MSE post. I hope that it is appropriate for this site.
Let $\sigma(x)$ be the sum of the divisors of $x$.
An integer $a$ is said to be solitary if ...

**2**

votes

**1**answer

447 views

### Is anything like $\phi(n)>\dfrac n{e^\gamma\log\log n},\ \sigma(n)<e^\gamma n\log\log n$ known/conjectured for the generalizations of these functions?

Is anything like $\dfrac n{\phi(n)}<\dfrac{\sigma(n)}n<e^\gamma\log\log n$ known/conjectured for the generalizations of these functions?
Let $n=p_1^{a_1}\cdots p_t^{a_t}$ be the canonical prime ...

**2**

votes

**1**answer

270 views

### Trying to prove a congruence for Stirling numbers of the second kind

This a repost of a question I asked at Stack Exchange, but I got no answer so far, so I am trying here, even though it may not suit the "research level" requirement.
Proposition: When $n$ and $m$ are ...

**2**

votes

**1**answer

154 views

### Cardinality of the prime divisor set of a k-power sum

Let $a_{1},\dots,a_{n}$ be positive natural numbers ($n>2$) such that $a_{i}\neq a_{j}$ if $i\neq j$. I want to prove that
$$ \left\lvert \left\{ p \text{ prime} \; : \; p \mid \sum_{i=1}^n a_{i}^{...

**7**

votes

**2**answers

708 views

### Numbers with all N-digit prefixes divisible by N

In base 10, the number 3816547290 contains every digit exactly once. When I take the first N digits, that substring is divisible by N. For example, 381 is divisible by 3, 38165 is divisible by 5, etc. ...

**0**

votes

**1**answer

483 views

### Is there a formula that can predict the primes in the sequence of ratios of consecutive superior highly composite numbers? : $2, 3, 2, 5, 2, 3, 7,…$

This is the sequence of prime numbers which are the elementary building blocks for the superior highly composite numbers:
$2, 3, 2, 5, 2, 3, 7, 2, 11, 13, 2, 3, 5, 17, 19, 2, 23, ...$
The $n^{th}$ ...

**34**

votes

**2**answers

1k views

### Does iterating a certain function related to the sums of divisors eventually always result in a prime value?

Let define the following function for integers (from 2): $f(x)=\sigma(x)-1$, where $\sigma$ is the sum of the divisors of $x$.
For example $f(6)=6+3+2=11$, $f(5)=5$.
Note that $x$ is a fixed point for ...

**5**

votes

**1**answer

520 views

### Arbitrarily large $n$ divides $F_n$

Is it true that there exists $n \in \mathbb{N}$ with arbitrarily many prime factors such that $n$ divides $F_n$, where $F_n$ represents the n-th Fibonacci number?

**10**

votes

**1**answer

293 views

### Laurent polynomials associated to partitions and a $Q$-deformation of $\sigma(d)$

Let $\alpha \vdash d$ be a partition of $d$, i.e. $\alpha = (\alpha_1 \geq \alpha_2 \geq …\geq \alpha_l)$, where $\sum_k \alpha_k = d$. Define a Laurent polynomial in $Q$ as follows:
$$
P_\alpha(Q) = ...

**2**

votes

**0**answers

288 views

### A question concerning the strange arithmetic derivation

This question is related to Strange (or stupid) arithmetic derivation. The original question whether an unbounded sequence of iterates exists is still unanswered.
$$n=\prod_{i=1}^{k}p_i^{\alpha_i} \...

**8**

votes

**2**answers

692 views

### runs of consecutive non squarefree integers

This question gained no attention at Math SE.
Call a sequence of $k$ consecutive naturals squary if each one of them is divided by a square > 1. The Chinese Remainder theorem trivially guarantees us ...

**5**

votes

**1**answer

336 views

### Who is attributed with the conjecture that every multiply-perfect number greater than $1$ is even?

I know that Descartes is considered to be the first to ask whether or not odd perfect numbers exist ($n$ such that $\sigma(n)=2n$, where $\sigma(n)$ is the sum of divisors of $n$), and he also ...

**11**

votes

**0**answers

325 views

### Do the coefficients of these irreducible polynomials always become periodic?

Fix $n\in\mathbb N$ and a starting polynomial (or seed) $p_n=a_0+a_1x+\dots+a_nx^n$ with $a_k\in\mathbb Z\ \forall k$ and $a_0a_n\ne0$.
Define $p_{n+1},p_{n+2},\dots$ recursively by $p_r = p_{r-1}+...

**5**

votes

**1**answer

210 views

### Large gaps between consecutive irreducible polynomials with small heights

For a prime gap of length at least $n$, a trivial upper bound for its first occurrence is $N=n!$ or $N=lcm(2,\dots,n)$. A bit better is $N=p_1\cdots p_n$ where $p_k$ is the $k$th prime, as then $N+2,\...

**6**

votes

**2**answers

713 views

### Sum of divisor function over arithmetic progression

I am trying to find an estimate for the following sum:
$$
\sum_{\substack{n \leq x \\ n \equiv k (m)}} d(n),
$$
where $d(n)$ is number of divisors of $n$. I found estimates for the case when $k$ and ...

**2**

votes

**0**answers

169 views

### Determine whether if $n$ is a primitive pseudoperfect (semiperfect) number, then $\sigma(n)<2^{\sigma_0(n)}$

Determine whether if $n$ is a primitive pseudoperfect (semiperfect) number, then $\sigma(n)<2^{\sigma_0(n)}$.
$\sigma_k(n)$ is the division function and $\sigma(n)=\sigma_1(n)$. A number is ...

**-1**

votes

**1**answer

203 views

### Are all known $k$-multiperfect numbers (for $k > 2$) not squarefree?

I asked the following question in MSE four ($4$) days ago, but so far nobody has posted an answer.
The gist of the question is as follows:
Are all known $k$-multiperfect numbers (for $k > ...

**2**

votes

**0**answers

213 views

### What are the divisors of $2n^2 - \sigma_{1}(n^2)$ for composite $n$?

What are the divisors of $2n^2 - \sigma_{1}(n^2)$ for composite $n$?
Here, $\sigma_{1}$ is the classical sum-of-divisors function. For example, $\sigma_{1}(3^2) = 1 + 3 + {3^2} = 13$.
(The function ...

**48**

votes

**4**answers

4k views

### Strange (or stupid) arithmetic derivation

Let us consider the following operation on positive integers: $$n=\prod_{i=1}^{k}p_i^{\alpha_i} \qquad f(n):= \prod_{i=1}^{k}\alpha_ip_i^{\alpha_i-1}$$ (Is it true that if we apply this operation to ...

**3**

votes

**1**answer

212 views

### Divisibility properties of a stream of numbers

Let $S$ be a set of $k$ distinct natural numbers, each
from the interval $[2,n]$, with least common multiple $\mathop{lcm}$.
What fraction $\rho$ of the numbers $2,3,4,\ldots,\mathop{lcm}$
are ...

**4**

votes

**0**answers

199 views

### Maximal order of Hooley's Delta function?

There is a large literature on Hooley's
$$
\Delta(n)=\max_u\sum_{d|n,\ e^u\le d< e^{u+1}}1
$$
giving its normal and average order. What is known of its maximal order?
Clearly $\Delta(n)\le d(n)$ ...

**3**

votes

**2**answers

1k views

### Finding primes using Euler's sum of divisors recurrence relation

Euler came up with following recurrence relation for the sum of divisors (refer to http://arxiv.org/abs/math/0411587)
$$\sigma(n) = \sigma(n−1) + \sigma(n−2) − \sigma(n−5) − \sigma(n−7) \dots$$
Since ...

**23**

votes

**2**answers

3k views

### Why do primes dislike dividing the sum of all the preceding primes?

I was investigating primes with the property that the sum of the first $n$ primes is divisible by $p_n$. It turns out that these primes are extremely extremely rare. For primes less than $10^9$, I ...

**3**

votes

**1**answer

303 views

### Ratio of consecutive divisors and average

Let $2\leq d_1 < d_2,...,d_l < n$ be all the proper nontrivial divisors of $n$. I like to understand how much these divisors deviates from each other. Here are two questions in this regard:
(1) ...

**2**

votes

**2**answers

413 views

### Is it necessary that gcd > 1 of an infinite set? [closed]

Consider an infinite set $S$, of positive integers.
If all the finite subsets of $S$ have GCD $>$ $1$, is it necessary that the GCD of $S$ is greater than $1$ as well?

**6**

votes

**6**answers

687 views

### 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 ...