All Questions
Tagged with divisor-sum sequences-and-series
37
questions
2
votes
0
answers
139
views
Divisors sum and Bessel Function related sums
Discovered the following relation:
$$\sum _{k=1}^{\infty } \sigma (k) \left(K_2\left(4 \pi \sqrt{k+y} \sqrt{y}\right)-K_0\left(4 \pi \sqrt{k+y} \sqrt{y}\right)\right)=\frac{\pi K_1(4 \pi y)-3 K_0(...
- 1,818
2
votes
1
answer
62
views
On the "efficiency" of digit-sum divisibility tricks in other bases (or about the growth rate of the number of divisors function).
Out of the different divisibility tricks there's a really simple rule that works for more than one divisor: The digit-sum. Specifically, if the sum of the digits of a number is a multiple of $1,3$ or $...
- 7,273
3
votes
1
answer
165
views
Why must $n-\lfloor\frac n2\rfloor+\lfloor\frac n3\rfloor-\dotsb$ grow like $n\ln2$? [duplicate]
Let
$$a(n):=n-\left\lfloor\frac n2\right\rfloor+\left\lfloor\frac n3\right\rfloor-\left\lfloor\frac n4\right\rfloor+\left\lfloor\frac n5\right\rfloor-\dotsb.$$
Note that this is a finite sum. Naïvely,
...
- 23.5k
1
vote
1
answer
66
views
$\sum_{k = 1}^{\infty} k\lfloor\frac{n}{k} \rfloor = 1 + \sum_{k = 1}^n \sigma_1(n)$
For any $f: \Bbb{N} \to \Bbb{Z}$ there exists a unique transformed function $F:\Bbb{N} \to \Bbb{Z}$ such that:
$$
f(n) = \sum_{k = 1}^{\infty}F_k\lfloor\frac{n}{k}\rfloor
$$
For example, set $F_1 = f(...
4
votes
1
answer
74
views
coefficient $[q^n]\sum_{m\ge1}\frac{q^{\alpha m}}{1-q^{\beta m}}$ where $0<\alpha<\beta$
Problem:
I am looking for a finite-sum expression for the coefficient $c_n=c_n(\alpha,\beta)$, where
$$C(\alpha,\beta;q)=\sum_{m\ge1}\frac{q^{\alpha m}}{1-q^{\beta m}}=\sum_{n\ge1}c_n(\alpha,\beta)q^n,...
- 17.3k
3
votes
2
answers
92
views
How to approach the negative root of the series $\sum\limits_{k=1}^{\infty} \sigma_1(k) x^k$
Let's define
$$
f(x) = \sum\limits_{k=1}^{\infty} \sigma_1(k) x^k, \quad \text{where } \sigma_1(k) = \sum\limits_{d|k}d
$$
Is there a way to approach the root $x_0$ of this series, where $-1 < x_0 &...
- 399
2
votes
1
answer
33
views
Convergente of sum of divisors sequence
Let $\sigma$ the application that transforms $n$ into the sum of its divisors (ex : $\sigma\left(6\right)=12$)\
I've proved that
$$
n+1 \leq \sigma\left(n\right) \leq n+n\ln\left(n\right)
$$
I know ...
- 7,367
0
votes
0
answers
37
views
Could we predict the exact value of $\sum_{n\geq 0}\dfrac{1}{\sigma(n!)}$ while it is convergent?
let $\sigma$ be sum of power divisor function , I'm interesting to evaluate the following sum $\sum_{n\geq 0}\dfrac{1}{\sigma(n!)}$ , we have that sum is converge using test creterion because we have ...
- 7,195
0
votes
1
answer
45
views
Confused as to how this step in a number theory proof is performed
How does this step
$$D(q)=\sum_{n=1}^\infty d(n)q^n$$
Become this step?
\begin{align}
D(q)
&=\sum_{n=1}^\infty\sum_{m|n}mq^n=\sum_{m=1}^\infty\sum_{m|n}mq^n \\
&=\sum_{m=1}^\infty\...
5
votes
0
answers
263
views
For what $k\in \mathbb{C}$ do we have $\lim_{x\to\infty}\frac{1}{x^{k+1}}\sum_{n=1}^x \sigma_k(n)=\frac{\zeta(k+1)}{k+1}$?
Can the following claim be extended for some complex $k$. Perhaps: all $k$ with real part greater than or equal to $1$? Do the arguments below fall apart for complex $k$ for some reason?
Claim. ...
- 3,813
0
votes
0
answers
89
views
The asymptotic behavior of $\sum_{n=1}^x\sigma_a(n)\sigma_b(n)$
Question
What is the asymptotic behavior of $$\sum_{n=1}^x\sigma_a(n)\sigma_b(n)$$
Where $\sigma_k=\sum_{d|n}d^k$
More generally I am curious if we can get bounds on $$\sum_{n=1}^x\prod_{i}\sigma_{...
- 3,813
0
votes
1
answer
163
views
On prime-perfect numbers and the equation $\frac{\varphi(n)}{n}=\frac{\varphi(\operatorname{rad}(n))}{\operatorname{rad}(\sigma(n))}$
While I was exploring equations involving multiple compositions of number theoretic functions that satisfy the sequence of even perfect numbers, I wondered next question (below in the Appendix I add a ...
user243301
3
votes
1
answer
106
views
A problem similar than that of the amicable pairs using the function $\operatorname{rad}(k)$: a first statement or conjecture
In this post we denote the product of distinct primes dividing an integer $k> 1$ as $\operatorname{rad}(k)$, with the definition $\operatorname{rad}(1)=1$, that is the so-called radical of an ...
user243301
6
votes
2
answers
226
views
On the equation involving the sum of divisors function $\sigma(105n+\sigma(n))=108\sigma(n)$
I am curious about the solutions of the following equation involving the sum of divisors function $\sigma(m)=\sum_{d\mid m}d$
$$\sigma(105n+\sigma(n))=108\sigma(n).\tag{1}$$
It is obvious, since $107$...
user243301
2
votes
1
answer
68
views
On miscellaneous questions about perfect numbers II
Let $\varphi(m)$ the Euler's totient function and $\sigma(m)$ the sum of divisors function. We also denote the product of primes dividing an integer $m>1$ as $\operatorname{rad}(m)$, that is the ...
user243301