Questions tagged [divisor-sum]
For questions on the divisor sum function and its generalizations.
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Define $\partial(n\mid x) = \partial(q_0 \cdots q_i \mid x) = \sum_{j=0}^i (-1)^i q_j (\frac{n}{q_j} \mid x)$. What does homology measure?
Let $R$ be a commutative ring with $1$ and let $M = \{ f : \Bbb{N} \to R \}$ be the $R$-module of arithmetic functions into $R$.
A basis for $M$ is $(d \mid \cdot) : d \in \Bbb{N}$ where $(d\mid n) = ...
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If $p$ is a prime number and $k$ is a positive integer, is it true that $\sigma_1(p^k) > 1 + k (\sqrt{p})^{1+k}$?
Denote the classical sum of divisors of the positive integer $x$ by $\sigma(x)=\sigma_1(x)$.
Here is my initial question:
If $p$ is a prime number and $k$ is a positive integer, is it true that
$$\...
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If $p^k m^2$ is an odd perfect number with special prime $p$ and $p = k$, then $\sigma(p^k)/2$ is not squarefree.
While researching the topic of odd perfect numbers, we came across the following implication, which we currently do not know how to prove:
CONJECTURE: If $p^k m^2$ is an odd perfect number with ...
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The mean square of $d_k(n)$
Let $d_2(n)=d(n)$ be the divisor function, and let $$d_k(n)=\sum_{d_1\cdots d_k= n}1=\sum_{m\cdot l= n}d_{k-1}(m).$$ Can anyone point me to a reference to the size of the error term when approximating
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On the prime factorization of $n$ and the quantity $J = \frac{n}{\gcd(n,\sigma(q^k)/2)}$, where $q^k n^2$ is an odd perfect number
Let $N = q^k n^2$ be an odd perfect number with special prime $q$ satisfying $q \equiv k \equiv 1 \pmod 4$ and $\gcd(q,n)=1$. Denote the classical sum of divisors of the positive integer $x$ by $\...
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On a consequence of $G \mid I \iff \gcd(G, I) = G$ (Re: Odd Perfect Numbers and GCDs)
Let $N = q^k n^2$ be an odd perfect number given in the so-called Eulerian form, where $q$ is the special prime satisfying $q \equiv k \equiv 1 \pmod 4$ and $\gcd(q,n)=1$. Denote the classical sum of ...
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Find all positive integers $n$ such that $\sigma(n) = n + 1 + \sigma(n+1)$
Find all positive integers $𝑛$ such that
$\sigma\left(n\right) = n + 1 + \sigma\left(n + 1\right)$, where $\sigma\,()$ is the divisor function.
I found $n = 18,\ 3200$.
For $n \leq 10^8$, only the ...
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Can this inequality involving the deficiency and sum of aliquot divisors be improved? - Part II
This MSE question (from April 2020) asked whether the inequality
$$\frac{D(n^2)}{s(n^2)} < \frac{D(n)}{s(n)}$$
could be improved, where $D(x)=2x-\sigma(x)$ is the deficiency of the positive integer ...
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Is $650$ the only solution not fitting in the family?
Inspired by this question
The linked question conjectures that $\frac{\sigma(n)}{n+1}$ (where $\sigma(n)$ denotes the divisor-sum function) is not an integer for any squarefree composite number.
If we ...
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On the equation $s(n^2) = \left(\frac{q-1}{2}\right)\cdot{D(n^2)}$, if $q^k n^2$ is an odd perfect number with special prime $q$
Let $N$ be an odd perfect number given in the so-called Eulerian form
$$N = q^k n^2$$
where $q$ is the special prime satisfying $q \equiv k \equiv 1 \pmod 4$ and $\gcd(q,n)=1$.
In what follows, let ...
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On Tony Kuria Kimani's recent preprint in ResearchGate
(Preamble: The method presented here to compute the GCD $g$ is patterned after the method used to compute a similar GCD in this answer to a closely related MSE question.)
Let $\sigma(x)=\sigma_1(x)$ ...
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Proof-verification request: On the equation $\gcd(n^2,\sigma(n^2))=D(n^2)/s(q^k)$ - Part II
(Preamble: This inquiry is an offshoot of this answer to a closely related question.)
In what follows, denote the classical sum of divisors of the positive integer $x$ by $\sigma(x)=\sigma_1(x)$, the ...
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If $2^r b^2$ is an even almost perfect number that is NOT a power of two, does it follow that $r=1$?
(The following are taken from this preprint by Antalan and Dris.)
Antalan and Tagle showed that an even almost perfect number $n \neq 2^t$ must necessarily have the form $2^r b^2$ where $r \geq 1$, $\...
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Does $\sigma(n^2)/q \mid q^k n^2$ imply $\sigma(n^2)/q \mid n^2$, if $q^k n^2$ is an odd perfect number with special prime $q$? - Part II
(Preamble: This inquiry is an offshoot of this MSE question.)
Denote the classical sum of divisors of the positive integer $x$ by $\sigma(x)=\sigma_1(x)$. Denote the abundancy index of $x$ as $I(x)=\...
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prime factorizations/ sum of squares of divisors.
Find all positive integers $n$ such that the sum of the squares of the divisors of $n$ is equal to $n^2+2n+37$, and in which $n$ is not of the form $p(p+6)$ where p and p+6 are prime numbers.
I tried ...
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