All Questions
Tagged with divisor-sum summation
45
questions
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
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
0
votes
1
answer
33
views
Question on an equation involving sum of a function over divisors. [closed]
I have a simple question regarding a particular form of a sum and I was hoping someone could provide some insights or guidance.
I was wondering if there was any other way to express the following sum ...
- 27
0
votes
2
answers
46
views
How to write this sum $\sum_{m=0}^\infty (2m+1)^{-2k-1} \sum_{r=1}^\infty (-1)^{r-1} e^{-2(2m+1)r a} $ as a sum over single index?
So I want to write the sum $$\sum_{m=0}^\infty (2m+1)^{-2k-1} \sum_{r=1}^\infty (-1)^{r-1} e^{-2(2m+1)r a} $$ where $a>0$ and $k\in \mathbb{N}$, as a sum over single index which probably uses odd ...
- 728
0
votes
0
answers
17
views
How can the following summation be written in terms of $N$?
Suppose that $N=\sum_{j=1}^{Q}k_j,$ where the $0\leq k_j$ are integers. Then, what is $\sum_{j=1}^{Q}jk_j$ in terms of $N$ and possibly $Q$. Meaning, if
$$f=\sum_{j=1}^{Q}jk_j,$$
what is either $f(N)$ ...
- 65
3
votes
1
answer
100
views
The summation $\sum_{n\geqslant1} \frac1n\sum_{d\mid n}\frac{d}{n^2+d}.$
I wish to evaluate $\sum\limits_{n\geqslant1}\frac1n\sum\limits_{d\mid n}\frac{d}{n^2+d}.$
Some observations:
Let $f(n)=\sum\limits_{d\mid n}\frac{d}{n^2+d}$.
Then $f(p)=\frac{p^2+p+2}{(p^2+1)(p+1)...
- 5,230
1
vote
0
answers
93
views
Simplify $\sum_{k = 1}^{\left\lfloor{N/2}\right\rfloor} \sum_{i = 1}^{k} \sum_{d_1, d_2 = 1}^{N} \left[{{d}_{1}\, {d}_{2} = i (2\, k - i)}\right]$
This restricted divisor problem is derived from counting the number of reducible quadratics where we have for one of the four main terms
$$\sum_{k = 1}^{\left\lfloor{N/2}\right\rfloor} \sum_{i = 1}^{...
- 967
1
vote
1
answer
75
views
What is meant by $\sum_{d \le x}f(d)$ in this equation?
Wikipedia's page (here) on the average order of arithmetic functions gives the following as a means of finding such an order using Dirichlet Series:
Define $f$ as an arithmetic function on $n$, and ...
2
votes
1
answer
72
views
Conjecture on the solutions to the equation $J(x) = J(x+a)$
Propose $J(x)$, which is a function that takes in a number and outputs the sum of all its factors (including itself)
Firstly, I think it's pretty interesting as it allows you to describe certain ...
- 446
1
vote
2
answers
107
views
Asymptotic expansion as $N \rightarrow \infty$ of $\sum_{k=1}^{\left\lfloor{N/2}\right\rfloor} k \sum_{e \mid 2k}\frac{\Lambda \left({e}\right)}{e}$
This expression comes from the asymptotic expansion of
$$\sum_{k=1}^{\left\lfloor{N/2}\right\rfloor} \sum_{i=1}^{k} \tau \left({i \left({2\, k - i}\right)}\right)$$
From Adrian W. Dudek, "Note on ...
- 967
3
votes
0
answers
77
views
Develop asymptotic as $N \rightarrow \infty$ of $\sum_{k=1}^{\left\lfloor{N/2}\right\rfloor} \sum_{i=1}^{k} \sum_{d \mid i (2k-i), d > N} 1$
This calculation arises from expansions of the number of factorable quadratics where the coefficients are constrained by naive height $N$. In the above formula $d$ are the divisors of $i \left({2\, k ...
- 967
1
vote
0
answers
53
views
PROOF: A Relationship Between A Natural Number and The Quantity of Its Divisors' Divisors
An arbitrary natural number $N \in \mathbb{N}$ has the divisors $d_1,d_2,...,d_s$ , where $N$ has $s$ divisors in total. If $n_i$ denotes the number of dividers to $d_i, i = 1, ..., s$, it can be ...
user736333
1
vote
0
answers
98
views
Prove that $\sum_{u = 1}^{N} \sum_{v=1}^{N} \left(\left[\sqrt{u^2-4v} \in Z\right]+\left[\sqrt{u^2+4v} \in Z\right]\right) = D \left(N\right)-1$
For a given height $N$. I am performing various sums involved in the counting of the number of monic quadratics, cubics, etc. that factor over a range. I noticed by experiment that the above sum ...
- 967
4
votes
1
answer
485
views
On the sum of sum of divisors $\sum_{a=1}^{N} D \left({\left\lfloor{\frac{N}{a}}\right\rfloor}\right)$.
where $D \left({x}\right)$ is the sum of divisors. This sum comes from my work on the number of reducible monic cubics. This is a two part question. By writing out all the divisors $\tau \left({a}\...
- 967
5
votes
0
answers
354
views
Fast Computation of the Divisor Summatory Function $\sum_{i=1}^x\sum_{d|i}d$
Denote by $\sigma_1(n)$ the sum of the divisors of $n$. For example, when $n=9$ we get $\sigma_1(9) = 1+3+9=13$. Define $D(x) = \sum_{n=1}^x\sigma_1(n)$. Varius methods are known for computing $D(x)$, ...
- 1,022