All Questions
Tagged with divisor-sum prime-numbers
72
questions
2
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55
views
The number of positive divisors of a number that are not present in another number.
How many positive divisors are there of $30^{2024}$ which are not divisors of $20^{2021}$?
I have tried many ways to try to get a pattern for this problem but I can't. I know that $30$ has $8$ ...
- 21
3
votes
2
answers
492
views
Square of prime numbers
This conjecture is inspired by the comment of @Eric Snyder: Prime numbers which end with 03, 23, 43, 63 or 83
$n$ is a natural number $>1$, $\varphi(n)$ denotes the Euler's totient function, $P_n$ ...
- 87
3
votes
0
answers
158
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Prime numbers which end with 03, 23, 43, 63 or 83
This is inspired from this post: Prime numbers which end with $19, 39, 59, 79$ or $99$
Here I found a new formula:
$n$ is a natural number $>1$, $\varphi(n)$ denotes the Euler's totient function, $...
- 87
-1
votes
1
answer
121
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Prime numbers which end with $59$ or $79$ [closed]
This is the weak conjecture of https://math.stackexchange.com/questions/4834936/prime-numbers-which-end-with-19-39-59-79-or-99.\
$\varphi(n)$ denotes the Euler’s totient function, $n$ denotes a ...
- 87
6
votes
1
answer
258
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A question about prime numbers, totient function $ \phi(n) $ and sum of divisors function $ \sigma(n) $
I noticed something interesting with the totient function $ \phi(n) $ and sum of divisors function $ \sigma(n) $ when $n > 1$.
It seems than :
$ \sigma(4n^2-1) \equiv 0 \pmod{\phi(2n^2)}$ only if ...
- 141
1
vote
1
answer
160
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Extraordinary Numbers
Can you please explain what are Extraordinary Numbers in detail? At the same time, I would also like to confirm whether the equivalent problem of Riemann Hypothesis mentioned here is correct (like it'...
- 157
1
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0
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53
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Define $\partial(n\mid x) = \partial(q_0 \cdots q_i \mid x) = \sum_{j=0}^i (-1)^i q_j (\frac{n}{q_j} \mid x)$. What does homology measure?
Let $R$ be a commutative ring with $1$ and let $M = \{ f : \Bbb{N} \to R \}$ be the $R$-module of arithmetic functions into $R$.
A basis for $M$ is $(d \mid \cdot) : d \in \Bbb{N}$ where $(d\mid n) = ...
0
votes
1
answer
94
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If $p$ is a prime number and $k$ is a positive integer, is it true that $\sigma_1(p^k) > 1 + k (\sqrt{p})^{1+k}$?
Denote the classical sum of divisors of the positive integer $x$ by $\sigma(x)=\sigma_1(x)$.
Here is my initial question:
If $p$ is a prime number and $k$ is a positive integer, is it true that
$$\...
- 8,010
0
votes
1
answer
339
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Finding a formula for $\sum_{d|n} \tau(d)$, where $\tau(d)$ is the number of divisors of $d$.
I am currently in the middle of the following exercise:
Exercise. Compute $$ \sum_{d|n} \left(\sigma(d)\mu\left(\frac{n}{d}\right)+\tau(d)\right),$$
where $\sigma$ is the function that corresponds to ...
- 1,141
1
vote
0
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53
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How can we show this relationship between the recursive sum of divisors function and Figurate Number Polynomial on Primes?
Let us define the following recursive function involving the sum of divisors function $\sigma(n)$:
\begin{array}{ l }
r(n,1)=\sigma(n) \\
r(n,2)=\sum_{d|n}r(d,1) \\
r(n,3)=\sum_{d|n}r(d,2) \\
\...
- 1,594
4
votes
2
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172
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Is $\limsup_n \frac{\sigma(n)}{n \log p(n)} <\infty$, where $p(n)$ is the greatest prime factor of $n$ and $\sigma(n)=\sum_{d | n} d$?
Let $\sigma(n)=\sum_{d | n} d$ and $p(n)$ be the greatest prime factor of $n$.
Can we prove that
$$\limsup_n \frac{\sigma(n)}{n \log p(n)} <\infty ?$$
I know that
$$\limsup_n \frac{\sigma(n)}{n \...
- 1,075
1
vote
1
answer
156
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Are there infinitely many primes $q$, such that $\sigma(p^k) = 2q$, where $p$ is prime and $p \equiv k \equiv 1 \pmod 4$?
Let $\sigma(x)=\sigma_1(x)$ be the classical sum of divisors of $x$. Denote the abundancy index of $x$ by $I(x)=\sigma(x)/x$.
Here is my question:
Are there infinitely many primes $q$, such that $\...
- 8,010
4
votes
2
answers
166
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The way to prove that $σ(a) = 3^k$ has no solution?
$\sigma(n)$ = sum of divisors of n is a divisors function.
How to prove there are no such $a$ and $k \ge 2$ satisfy $\sigma(a) = 3^k$.
This proplem can be simplify to the case when $a$ is a power of ...
- 69
1
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1
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71
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For sum of divisors of 4n+1 and 4n+3 and is there a known inequality with n?
I was trying to understand the relationship between $S(n)$ and $S(4n+1)$ and between $S(n)$ and $S(4n+3)$.
It seems that up to $10^5$, the ratios $S(4n+1)/S(n)$ and $S(4n+3)/S(n)$ are always $<1$, ...
- 59
3
votes
2
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615
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There is only one positive integer that is both the product and sum of all its proper positive divisors, and that number is $6$.
Confused as to how to show the number 6's uniqueness. This theorem/problem comes from the projects section of "Reading, writing, and proving" from Springer.
Definition 1. The sum of divisors ...
- 147