All Questions
Tagged with divisor-sum elementary-number-theory
395
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Does the following GCD divisibility constraint imply that $\sigma(m^2)/p^k \mid m$, if $p^k m^2$ is an odd perfect number with special prime $p$?
The topic of odd perfect numbers likely needs no introduction.
In what follows, denote the classical sum of divisors of the positive integer $x$ by $$\sigma(x)=\sigma_1(x).$$
Let $p^k m^2$ be an odd ...
- 8,010
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Improving $I(m^2)/I(m) < 2^{\log(13/12)/\log(13/9)}$ where $p^k m^2$ is an odd perfect number with special prime $p$
In what follows, let $I(x)=\sigma(x)/x$ denote the abundancy index of the positive integer $x$, where $\sigma(x)=\sigma_1(x)$ is the classical sum of divisors of $x$.
The following is an attempt to ...
- 8,010
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estimating an elementary sum involving divisor function
Please guide me as to how to obtain the below bound and whether it is optimal.
Let a squarefree integer $N=\prod_{1 \leq i \leq m} p_i$ be a product of $m$ primes ($p_1 < p_2 < \dots < p_m$) ...
- 417
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Efficient proof that a number is NOT a Zumkeller number?
The subset sum problem is known to be NP-complete , so in general there is no efficient method to decide it , in particular to prove a negative result.
This problem arises in the problem to decide ...
- 85.1k
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Exercise 1 Section 9.2. Montgomery/Vaughan's Multiplicative NT
I am trying to show that for any integer $a$, $$e(a/q) =
\sum_{d|q, d|a} \dfrac{1}{ϕ(q/d)} \sum_{χ \ (mod \ q/d)} χ(a/d) τ(χ).$$ First I considered the case $(a,q)=1$ and the mentioned equality holds. ...
- 281
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Can we efficiently check whether a number is a Zumkeller number?
A positive integer $n$ is a Zumkeller number iff its divisors can be partitioned into two sets with equal sum.
If $\sigma(n)$ denotes the divisor-sum-function , this means that there are distinct ...
- 85.1k
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Asymptotic for $\sum_{d\mid N}\frac{d^{2}}{\sigma\left(d\right)}$ [closed]
Let $N\in\mathbb{N}$. I'm looking for an asymptotic formula for $\sum_{d\mid N}\frac{d^{2}}{\sigma\left(d\right)}$ as $N\rightarrow\infty$. I tried to use known asymptotic formulas for similar ...
- 49
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Proof for the sum and number of positive divisors for a positive integer $n$. [duplicate]
I know that the number of positive divisors of $n$ can be given by :
$\tau(n)$ = $(a_1+1)(a_2+1)\ldots(a_k+1)$ where $n = p_1^{a_1}p_2^{a_2}.... p_k^{a_k}$, where $p_1, p_2... p_k$ are the prime ...
- 49
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Solutions of equations of form $\tau(n+a_i)=\tau(x+a_i)$ where $\tau$ is the number of divisors function and $a_i$ is a diverging series
Consider the function $$\tau(n)=\sum_{d|n}1$$ which gives the number of divisors of a number. The question is: How much information does $\tau(n)$ contain about $n$?
The answer is obviously: not very ...
- 103
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Growth rate of sum of divisors cubed [closed]
I a trying to find a result similar to:
$$\limsup_{n \to \infty} \frac{\sigma_1(n)}{n \log \log (n)} = e^\gamma$$
(where $\sigma_1$ is the sum of divisors function) but regarding the growth rate of $\...
- 1,919
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Determine all positive integers $n$ such that: $n+d(n)+d(d(n))+\dotsb=2023$.
For a positive integer number $n>1$, we say that $d(n)$ is its superdivisor if $d(n)$ is the largest divisor of $n$ such that $d(n)<n$. Additionally, we define $d(0)=d(1)=0$. Determine all ...
- 418
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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
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$\sum_{k = 1}^{\infty} k\lfloor\frac{n}{k} \rfloor = 1 + \sum_{k = 1}^n \sigma_1(n)$
For any $f: \Bbb{N} \to \Bbb{Z}$ there exists a unique transformed function $F:\Bbb{N} \to \Bbb{Z}$ such that:
$$
f(n) = \sum_{k = 1}^{\infty}F_k\lfloor\frac{n}{k}\rfloor
$$
For example, set $F_1 = f(...
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Prove that there are infinitely many natural number such that $σ(n)>100n$
The problem is as follows: Prove that there are infinitely many natural numbers such that $σ(n)>100n$. $σ(n)$ is the sum of all natural divisors of $n$ (e.g. $σ(6)=1+2+3+6=12$).
I have come to the ...
- 19
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Finding natural numbers with $12$ divisors $1=d_1<d_2<\cdots<d_{12}=n$, such that the divisor with the index $d_4$ is equal to $1+(d_1+d_2+d_4)d_8$.
Find the natural number(s) n with $12$ divisors $1=d_1<d_2<...<d_{12}=n$ such that the divisor with the index $d_4$, i.e, $d_{d_4}$ is equal to $1+(d_1+d_2+d_4)d_8$.
My work:
$$\begin{align}
...
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