All Questions
117
questions
0
votes
0
answers
81
views
Looking for a solution of $\sum_{i = 1}^{k} \sum_{{d}_{1}\, {d}_{2} = i (2k - i), {d}_{1} \le N, {d}_{2} \le N} [GCD(2 k, {d}_{1}, {d}_{2}) = 1]$
The double sum is
$$\sum_{i = 1}^{k} \sum_{\substack{{d}_{1}\, {d}_{2} = i \left({2k - i}\right), \\ {d}_{1} \le N, {d}_{2} \le N}} \left[{\left({2\, k, {d}_{1}, {d}_{2}}\right) = 1}\right]$$
where [.....
- 967
0
votes
1
answer
66
views
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
0
votes
2
answers
62
views
Comparing integral with a sum
Show that \begin{equation}\sum_{m=1}^k\frac{1}{m}>\log k.\end{equation}
My intuition here is that the LHS looks a lot like $\int_1^k\frac{1}{x}\textrm{d}x$, and this evaluates to $\log k$. To ...
- 135
1
vote
1
answer
69
views
Summation notation over divisors confusion
What does the following summation notation represent?
$\sum\limits_{d_1 \mid a, \; d_2\mid b}f(d_1d_2)=\sum\limits_{d_1\mid a }\sum\limits_{d_2 \mid b}f(d_1)f(d_1)=\sum\limits_{d_1\mid a}f(d_1)\sum\...
- 637
5
votes
2
answers
262
views
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
2
votes
0
answers
49
views
Weird computation on the (variant) divisor problem
I have a weird computations here about the (variation of the) divisor problems that involves the squarefull numbers. It is the problem to determine
$\displaystyle \sum_{a^2b^3\le x} 1,$ which is ...
- 1,033
0
votes
2
answers
85
views
Construct a prime using $[2, 2, 2, ...., 3]$
Is there a generalized method to constructing primes through sums using the set $[2, 2, 2, ..., 3]$ given its elements are $n$- many 2s and a 3. This question obviously requires knowledge on ...
user1107557
2
votes
0
answers
71
views
When is $\sum_{1 \leq n \leq k}n^{-n+k}$ prime?
Consider the following finite sum $f: \mathbb{N} \rightarrow \mathbb{N}$ defined as
$$f(k) = \sum_{1 \leq n \leq k}n^{-n+k}$$
$$ = 2 + 2^{k-2} + 3^{k-3}...+ (k-1)$$
It is easy to see that $f(2) = 2$ ...
user1107557
1
vote
1
answer
314
views
Number theory approach to Project Euler's "Large Sum" problem?
I am refreshing some of my skills by solving problems on the Project Euler site. It is a repository of problems that usually require some mathematics knowledge and programming knowledge to solve ...
- 1,876
1
vote
0
answers
85
views
Probability that two integers are coprime
Maybe it is a silly question, but anyway. We know that the probability that two positive integers are coprime is $6/\pi^2$. However, for fixed positive integers $r$ and $s$, I'd want to compute the ...
- 73
1
vote
2
answers
83
views
Find all the integers which are of form $\dfrac{b+c}{a}+\dfrac{c+a}{b}+\dfrac{a+b}{c}, a,b,c\in \mathbb{N}$, any two of $a,b,c$ are relatively prime.
I have a question which askes to find all the integers which can be
expressed as
$\displaystyle \tag*{} \dfrac{b+c}{a}+\dfrac{c+a}{b}+\dfrac{a+b}{c}$
where $a,b,c\in \mathbb{N} $ and any two of $a,b,...
- 921
4
votes
1
answer
301
views
Is this a known result on graph products?
Consider two undirected graphs $G=(V,E)$ and $H=(I,F)$.
Denote by $\mathcal N_G(v)$ (resp., $\mathcal N_{H}(i)$) the first neighborhood of a node $v\in V$ (resp., $i\in I$), including $v$ (resp., $i$)....
- 434
1
vote
1
answer
79
views
Can I factorize a double sum into a product?
Fix two positive constants $A,B>0$, two finite sets $\mathcal A, \mathcal B$, and two functions $\alpha,\beta \colon \mathcal A \times \mathcal B \to [0,1]$.
Assume that:
For all $b\in \mathcal B$,...
- 434
1
vote
1
answer
158
views
Maximum sum of digits from a given formula
The sum of digits of the telephone number aaabbbb equals the two-digit number ab.
What is the sum $a+b?$
$(A) \space8$
$(B)\space 9 $
$(C) \space10 $
$(D)\space 11$
$(E) \space 12$
So far I have tried ...
- 23
1
vote
1
answer
183
views
How summation is changed in Analytic number theory
Consider this expression S(x, z) = $\sum_{n\leq x} \sum_{{d|n , d|P(z) } }\mu(d) $ . I don't understand the logic behind next step and get really confused on how summation is changed.
In next step ...
user775699