Questions tagged [divisor-sum]
For questions on the divisor sum function and its generalizations.
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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 ...
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Is there any square harmonic divisor number greater than $1$?
A harmonic divisor number or Ore number is a positive integer whose harmonic mean of its divisors is an integer. In other words, $n$ is a harmonic divisor number if and only if $\dfrac{nd(n)}{\sigma(n)...
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Extraordinary Numbers
Can you please explain what are Extraordinary Numbers in detail? At the same time, I would also like to confirm whether the equivalent problem of Riemann Hypothesis mentioned here is correct (like it'...
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$\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 ...
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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:
$$\begin{align}
...
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Question on an equation involving sum of a function over divisors. [closed]
I have a simple question regarding a particular form of a sum and I was hoping someone could provide some insights or guidance.
I was wondering if there was any other way to express the following sum ...
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Equality of two sums involving hecke eigenvalues in a paper of Luo and Sarnak
I am reading the paper Mass Equidistribution for Hecke Eigenforms by Luo and Sarnak. In the paper there is the following equality:
By the multiplicativity of Hecke eigenvalues, we have
$$ \sum_{r\geq ...
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Continued aliquot sums
What happens if one takes the aliquot sum of an integer and then repeats the process so that one takes the aliquot sum of all of those factors that were not reduced to the number 1 on the previous ...
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Limit of convolution sums of divisor functions
In this paper, Ramanujan studies the convolution sum of divisor functions, which he denotes as $$\sum_{r,s}(n) := \sum_{m = 0}^n \sigma_r(m) \sigma_s(n-m),$$ where above, he defines $\sigma_s(0) = \...
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On a conjecture involving multiplicative functions and the integers $1836$ and $137$
We denote the Euler's totient as $\varphi(x)$, the Dedekind psi function as $\psi(x)$ and the sum of divisors function as $\sigma(x)$. Are well-known arithmetic functions, see the corresponding ...
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On a conjecture involving multiplicative functions and the integers $1836$ and $136$
We denote the Euler's totient as $\varphi(x)$, the Dedekind psi function as $\psi(x)$ and the sum of divisors function as $\sigma(x)$. Are well-known arithmetic functions, see Wikipedia. I would like ...
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Who discovered the largest known $3$-perfect number in $1643$?
Multiperfect numbers probably need no introduction. (These numbers are defined in Wikipedia and MathWorld.)
I need the answer to the following question as additional context for a research article ...
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Is this a new representation of (some) Bernoulli numbers?
Let $\operatorname{B}(n)$ denote the Bernoulli numbers and
$\operatorname{b}(n) = \operatorname{B}(n)/n$ with $b(0)=1$
the divided Bernoulli numbers. Also let $\sigma_{k}(n)= \sum_{d \mid n} d^k$
...
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Does an odd perfect number have a divisor (other than $1$) which must necessarily be almost perfect?
Let $\sigma(x)=\sigma_1(x)$ be the classical sum of divisors of the positive integer $x$. Denote the aliquot sum of $x$ by $s(x)=\sigma(x)-x$ and the deficiency of $x$ by $d(x)=2x-\sigma(x)$. ...
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