All Questions
Tagged with divisor-sum upper-lower-bounds
67
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Improving $I(m^2)/I(m) < 2^{\log(13/12)/\log(13/9)}$ where $p^k m^2$ is an odd perfect number with special prime $p$
In what follows, let $I(x)=\sigma(x)/x$ denote the abundancy index of the positive integer $x$, where $\sigma(x)=\sigma_1(x)$ is the classical sum of divisors of $x$.
The following is an attempt to ...
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Divisors of $x^2-1$ in Brocard's Problem
In this post, I was curious if the divisor bound could be improved for the product of two consecutive even numbers. It seems most likely it cannot by much. How could the upper bound of $\sigma_0(x^2 - ...
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Maxima of the Sum of Divisors Function and Upper Bound of a similar Ratio
If we define a function (aka Gronwall’s function) as: $$F(n)=\frac{\sigma(n)}{n \log \log n}$$
Then for $n>15$, it does have an upper bound. I want to know what's that specific upper bound is?
Also ...
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Is there an analytical solution to the inequality $\frac{1 + k({p}^{(k+1)/2)})}{p^k} < \frac{p}{p - 1}$ if one were to bound $k$ in terms of $p$?
My question is as is in the title:
Is there an analytical solution to the inequality
$$\frac{1 + k({p}^{(k+1)/2)})}{p^k} < \frac{p}{p - 1}$$ if one were to bound $k$ in terms of $p$?
Here, $p \...
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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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Does the following lower bound improve on $I(q^k)+I(n^2) > \frac{57}{20}$, where $q^k n^2$ is an odd perfect number?
Preamble: This question is an offshoot of this earlier post. (This inquiry has likewise been cross-posted to MO last June $10, 2022$.)
Let $N = q^k n^2$ be an odd perfect number with special prime $q$...
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Why is this alternating summation involving reciprocals of divisors always positive? A conjecture.
Conjecture. For all $n,m \in \Bbb{N}$,
$$
f(n, m) := \sum_{c\mid d\mid n\# \\ \gcd(c, 2m) = 1}\dfrac{(-1)^{\omega(d)}}{d}
$$
is greater than $0$.
Example verification code:
...
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Does $\frac{\sigma(q^k)}{n} + \frac{\sigma(n)}{q^k} < 10$ hold in general for an odd perfect number $q^k n^2$ with special prime $q$?
This question is an offshoot of this MSE answer.
Let $\sigma(x)$ be the sum of the divisors of the positive integer $x$. (Denote the abundancy index of $x$ by $I(x)=\sigma(x)/x$.)
If $\sigma(M) = 2M$...
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Does the inequality $I(n^2) \leq 2 - \frac{5}{3q}$ improve $I(q^k) + I(n^2) < 3$, if $q^k n^2$ is an odd perfect number with special prime $q$?
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 abundancy index of the positive integer $x$ by $I(x)=\sigma(x)/...
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Bounds for the abundancy index of divisors of odd perfect numbers in terms of the deficiency function - Part II
This post is an offshoot of this MSE question.
Motivation
Let $x, y$ and $z$ be positive integers.
Denote the sum of divisors of $x$ by $\sigma(x)$. Also, denote the deficiency of $y$ by $D(y)=2y-\...
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On the quantity $I(q^k) + I(n^2)$ considered as functions of $q$ and $k$, when $q^k n^2$ is an odd perfect number with special prime $q$ - Part II
Hereinafter, let $N = q^k n^2$ be an odd perfect number with special/Euler prime $q$ satisfying $q \equiv k \equiv 1 \pmod 4$ and $\gcd(q,n)=1$.
Denote the abundancy index of the positive integer $x$ ...
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On the conjectured inequality $q > k$, where $q^k n^2$ is an odd perfect number with special prime $q$
Let $N=q^k n^2$ be an odd perfect number with special/Eulerian prime $q$ satisfying $q \equiv k \equiv 1 \pmod 4$ and $\gcd(q,n)=1$.
Descartes, Frenicle, and subsequently Sorli conjectured that $k=1$ ...
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On the inequality $I(q^k)+I(n^2) \leq \frac{3q^{2k} + 2q^k + 1}{q^k (q^k + 1)}$ where $q^k n^2$ is an odd perfect number
(Note: This post is an offshoot of this earlier MSE question.)
Let $N = q^k n^2$ be an odd perfect number with special prime $q$ satisfying $q \equiv k \equiv 1 \pmod 4$ and $\gcd(q,n)=1$.
Define the ...
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Is it possible to improve on these bounds for $\frac{\varphi(n)}{n}$, if $q^k n^2$ is an odd perfect number with special prime $q$?
Let $N = q^k n^2$ be an odd perfect number with special/Eulerian prime $q$ satisfying $q \equiv k \equiv 1 \pmod 4$ and $\gcd(q,n)=1$. Denote the Euler-totient function of the positive integer $x$ by ...
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A proposed unconditional factor-chain proof for the inequality $I(n) > \frac{3}{2}$, where $q^k n^2$ is an odd perfect number with special prime $q$
Let $N = q^k n^2$ be a hypothetical odd perfect number given in Eulerian form. (That is, $q$ is the special/Euler prime satisfying $q \equiv k \equiv 1 \pmod 4$ and $\gcd(q,n)=1$.)
Denote the ...
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