All Questions
Tagged with divisor-sum inequality
87
questions
2
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2
answers
183
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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 $d(x)=2x-\sigma(x)$. ...
- 8,010
0
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2
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102
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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 \...
- 8,010
0
votes
1
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94
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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
$$\...
- 8,010
1
vote
1
answer
223
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Can this inequality involving the deficiency and sum of aliquot divisors be improved? - Part II
This MSE question (from April 2020) asked whether the inequality
$$\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 ...
- 8,010
0
votes
4
answers
100
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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$...
- 8,010
0
votes
3
answers
193
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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$ ...
- 8,010
1
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1
answer
115
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If $N = q^k n^2$ is an odd perfect number with special prime $q$, then must $\sigma(n^2)$ be abundant?
Preamble: This post is an offshoot of this earlier MSE question.
The topic of odd perfect numbers likely needs no introduction.
Let $\sigma=\sigma_{1}$ denote the classical sum of divisors. Denote ...
- 8,010
1
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0
answers
76
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On the sum $I(q^k) + I(n^2)$, where $q^k n^2$ is an odd perfect number with special prime $q$
Denote the classical sum of divisors of the positive integer $x$ by $\sigma(x)=\sigma_1(x)$, and denote the abundancy index of $x$ by $I(x)=\sigma(x)/x$.
Let $N = q^k n^2$ be an odd perfect number ...
- 8,010
1
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0
answers
37
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Does this "improvement" to $I(m) < 2$, if $p^k m^2$ is an odd perfect number with special prime $p$, work?
(Preamble: This question is an offshoot of this earlier post.)
Denote the classical sum of divisors of the positive integer $x$ by $\sigma(x)=\sigma_1(x)$, and the abundancy index of $x$ by $I(x)=\...
- 8,010
2
votes
2
answers
217
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If $p^k m^2$ is an odd perfect number with special prime $p$, then which is larger: $D(p^k)$ or $D(m)$?
(Preamble: This question is tangentially related to this earlier post.)
Denote the classical sum of divisors of the positive integer $x$ to be $\sigma(x)=\sigma_1(x)$. Denote the abundancy index of $...
- 8,010
2
votes
1
answer
133
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If $p^k m^2$ is an odd perfect number with special prime $p$, then which is larger: $D(p^k)$ or $D(m^2)$?
Denote the classical sum of divisors of the positive integer $x$ to be $\sigma(x)=\sigma_1(x)$. Denote the abundancy index of $x$ by $I(x)=\sigma(x)/x$. Finally, denote the deficiency of $x$ by $D(x)...
- 8,010
0
votes
1
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189
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Why does an odd perfect number seemingly "violate" basic inequality rules?
Let $p^k m^2$ be an odd perfect number (OPN) with special prime $p$ satisfying $p \equiv k \equiv 1 \pmod 4$ and $\gcd(p,m)=1$.
My question is as is in the title:
Why does an OPN seemingly "...
- 8,010
1
vote
2
answers
116
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On a curious equation regarding odd perfect numbers
Let $p^k m^2$ be an odd perfect number (OPN) with special prime $p$ satisfying $p \equiv k \equiv 1 \pmod 4$ and $\gcd(p,m)=1$.
Since $\gcd(p^k, \sigma(p^k))=1$, then we essentially get the equation
$$...
- 8,010
1
vote
3
answers
238
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If $p^k m^2$ is an odd perfect number, then what is the optimal constant $C$ such that $\frac{\sigma(m^2)}{p^k} < \frac{m^2 - p^k}{C}$?
The topic of odd perfect numbers likely needs no introduction.
The question is as is in the title:
If $p^k m^2$ is an odd perfect number with special prime $p$, then what is the optimal constant $C$ ...
- 8,010
2
votes
1
answer
46
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Does this inequality hold for all odd $n$: $\sigma_{n-1}\sigma_{n+3} > \sigma_{n}\sigma_{n+2}$?
Let be $\sigma_n = \sum\limits_{d|n}d$ and let $n$ be an odd number, then the following inequality holds numerically up to $n = 10^7$.
$$
\begin{align}
\sigma_{n-1}\sigma_{n+3} > \sigma_{n}\sigma_{...
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