All Questions
Tagged with arithmetic-functions perfect-numbers
92 questions
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Does the following GCD divisibility constraint imply that $\sigma(m^2)/p^k \mid m$, if $p^k m^2$ is an odd perfect number with special prime $p$?
The topic of odd perfect numbers likely needs no introduction.
In what follows, denote the classical sum of divisors of the positive integer $x$ by $$\sigma(x)=\sigma_1(x).$$
Let $p^k m^2$ be an odd ...
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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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2
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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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Is this disproof for the Descartes-Frenicle-Sorli Conjecture that $k=1$, if $p^k m^2$ is an odd perfect number, valid?
Let $N = p^k m^2$ be an odd perfect number with special prime $p$ satisfying $p \equiv k \equiv 1 \pmod 4$ and $\gcd(p,m)=1$.
It is known that
$$D(p^k)D(m^2)=2s(p^k)s(m^2) \tag{0}$$
where $D(x)=2x-\...
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Why is it that, if there are no odd perfect numbers, then there are no other $3$-perfect numbers, apart from the six known, as of the year $1643$?
Let $\sigma(x)=\sigma_1(x)$ denote the classical sum of divisors of the positive integer $x$.
A number $N$ is said to be $k$-perfect if $\sigma(N)=kN$ where $k$ is a positive integer.
The number $1$ ...
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1
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70
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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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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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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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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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Are these valid proofs for the equation $\gcd(n,\sigma(n^2))=\gcd(n^2,\sigma(n^2))$, 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 $q \equiv k \equiv 1 \pmod 4$.
It is known that
$$i(q)=\gcd(n^2,\sigma(n^2))=\frac{\sigma(n^2)}{q^k}=\frac{n^2}{\sigma(q^k)/...
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1
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Are the (Bezout) coefficients for $\gcd(n^2, \sigma(n^2)) = q\sigma(n^2) - 2(q - 1)n^2$ (where $q^k n^2$ is an odd perfect number) unique?
In what follows, denote the classical sum of divisors of the positive integer $x$ by $\sigma(x)=\sigma_1(x)$, the deficiency of $x$ by $D(x)=2x-\sigma(x)$, and the aliquot sum of $x$ by $s(x)=\sigma(x)...
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Is the argument used in this proof that $k=1$ logically sound, where $q^k n^2$ is an odd perfect number with special prime $q$?
The topic of odd perfect numbers likely needs no introduction.
Euler proved that a hypothetical odd perfect number $N$, if one exists, must have the so-called Eulerian form $N=q^k n^2$, where $q$ is ...
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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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Follow-up to MSE question 3738458
This is a follow-up inquiry to this MSE question.
Let $\sigma(x)$ denote the sum of divisors of the positive integer $x$.
A number $M$ is said to be perfect if $\sigma(M)=2M$. For example, $6$ and $...
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What conditions on $X$ will guarantee that $\gcd(\text{square part of } X,\text{squarefree part of } X)=1$, if $X$ is neither a square nor squarefree?
The following query is an offshoot of this answer to a closely related post.
Denote the classical sum of divisors of the positive integer $x$ by $\sigma(x)=\sigma_1(x)$.
The topic of odd perfect ...
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Does $I = \gcd(n,\sigma(n^2)) = (\frac{n}{\sigma(q^k)/2})\cdot\gcd(\sigma(q^k)/2,n)$ imply that $\sigma(q^k)/2 \mid n$ holds?
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 GCDs:
$$G = \gcd\bigg(\sigma(q^k),\sigma(n^2)\bigg)$$
$$H = \...
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2
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Does $\sigma(m^2)/p^k$ divide $m^2 - p^k$, if $p^k m^2$ is an odd perfect number with special prime $p$?
The following query is an offshoot of this post 1 and this post 2.
Denote the classical sum of divisors of the positive integer $x$ by $\sigma(x)=\sigma_1(x)$.
The topic of odd perfect numbers likely ...
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2
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327
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What are the remaining cases to consider for this problem, specifically all the possible premises for $i(q)$?
Denote the classical sum of divisors of the positive integer $x$ by $\sigma(x)=\sigma_1(x)$. (Note that the divisor sum $\sigma$ is a multiplicative function.)
A number $P$ is said to be perfect if $\...
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4
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102
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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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On the reciprocal of $I(n^2) - \frac{2(q - 1)}{q}$, if $q^k n^2$ is a(n) (odd) perfect number with special prime $q$
Denote the classical sum of divisors of the positive integer $x$ by $\sigma(x)=\sigma_1(x)$. Denote the abundancy index of $x$ by $I(x)=\sigma(x)/x$.
A number $N$ is said to be perfect if $\sigma(N)=...
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3
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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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1
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1
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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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If $\gcd(n^2, \sigma(n^2)) = q\sigma(n^2) - 2(q - 1) n^2$, does it follow that $q n^2$ is perfect?
Let $q$ be an (odd) prime, and let $\gcd(q,n)=1$.
Denote the classical sum of divisors of the positive integer $x$ by $\sigma(x)=\sigma_1(x)$.
A number $N$ is said to be perfect if $\sigma(N)=2N$.
...
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1
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1
answer
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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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2
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Help with "A Simpler Dense Proof regarding the Abundancy Index."
I'm reading Richard Ryan's article "A Simpler Dense Proof regarding the Abundancy Index" and got stuck in his proof for Theorem 2. The Theorem is stated as follows:
Suppose we have a ...
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If $k=1$, then the index of an odd perfect number $p^k n^2$ (as defined by Chen and Chen (2014)) is not squarefree.
In what follows, we let $\sigma=\sigma_1$ denote the classical sum-of-divisors function. Denote the deficiency function of the positive integer $x$ by $D(x)=2(x)-\sigma(x)$, and the aliquot sum of $x$...
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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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1
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On odd perfect numbers $q^k n^2$ and the deficient-perfect divisor $q^{\frac{k-1}{2}} n^2$ - Part III
(Preamble: This question is an offshoot of this answer by mathlove to an earlier post.)
Let $\sigma(x)=\sigma_1(x)$ denote the classical sum of the divisors of the positive integer $x$.
If $\sigma(m)=...
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2
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2
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97
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On improving the upper bound $I(m^2) \leq \frac{2p}{p+1}$, if $p^k m^2$ is an odd perfect number with special prime $p$
(Preamble: This question is an offshoot of this earlier MSE post.)
Let $p^k m^2$ be an odd perfect number (OPN) with special prime $p$ satisfying $p \equiv k \equiv 1 \pmod 4$ and $\gcd(p,m)=1$.
...
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110
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Will it be possible to compute a factored expression for $n^2 - q^k$, if $q^k n^2$ is an odd perfect number with special prime $q$?
In what follows, we denote the classical sum of divisors of the positive integer $x$ by
$$\sigma(x)=\sigma_1(x)=\sum_{d \mid x}{d},$$
and the abundancy index of $x$ by $I(x)=\sigma(x)/x$.
If $N$ is ...
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1
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1
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119
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On the inequality $I(n^2) \geq (6 - s(q^k))/3$, where $q^k n^2$ is an odd perfect number and $I(x)$ is the abundancy index of $x$
In what follows, we denote the classical sum of divisors of the positive integer $x$ by $\sigma(x)=\sigma_1(x)$, the deficiency of $x$ by $D(x)=2x-\sigma(x)$, the aliquot sum of $x$ by $s(x)=\sigma(x)-...
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71
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On an inequality involving the abundancy index of the Eulerian component $p^k$ of an odd perfect number $p^k m^2$ - Part II
Preamble: This post is an offshoot of this earlier question.
The topic of odd perfect numbers likely needs no introduction.
Denote the classical sum of divisors of the positive integer $x$ by $\sigma(...
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1
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1
answer
124
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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 ...
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Proof-verification request: If $q^k n^2$ is an odd perfect number with special prime $q$, then $q < n^{2/5}$.
Hereinafter, we will denote the classical sum of divisors of the positive integer $x$ by $\sigma(x)=\sigma_1(x)$.
The topic of odd perfect numbers likely needs no introduction.
Let $q^k n^2$ be an odd ...
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2
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2
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182
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Is this proof regarding the nonexistence of odd perfect numbers correct?
Preamble: This post is an offshoot of this earlier question, which was not so well-received in MathOverflow.
Let $\sigma(x)=\sigma_1(x)$ denote the classical sum of divisors of the positive integer $...
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2
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1
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Is it true that $\frac{D(n^2)}{s(q^k)} > \frac{\sigma(q^k)}{2}$, if $q^k n^2$ is an odd perfect number given in Eulerian form?
Preamble: This post is an attempt to resolve the problem in this earlier question.
Let $\sigma(x)$ denote the sum of divisors of the positive integer $x$.
A number $M$ is said to be perfect if $\...
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2
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1
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73
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On the equation $\frac{1}{q}=1-\frac{\varphi(N)/N}{\varphi(n)/n}$, where $N=q^k n^2$ is an odd perfect number with special prime $q$
Preamble: This question is an offshoot of this earlier MSE post.
Denote the classical sum of divisors of the positive integer $x$ by $\sigma(x)=\sigma_1(x)$, and the Euler totient function of $x$ by $\...
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0
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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 ...
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1
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1
answer
127
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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 $q$?
Denote the classical sum of divisors of the positive integer $x$ by $\sigma(x)=\sigma_1(x)$. Denote the abundancy index of $x$ as $I(x)=\sigma(x)/x$.
I discovered an interesting identity involving ...
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3
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525
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Does the following lower bound improve on $I(q^k) + I(n^2) > 3 - \frac{q-2}{q(q-1)}$, where $q^k n^2$ is an odd perfect number? - Part II
Preamble: This question is an offshoot of these earlier posts: (post1), (post2).
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,...
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2
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2
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220
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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 $...
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2
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0
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50
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Improving $I(m) < 2$, if $p^k m^2$ is an odd perfect number with special prime $p$
The topic of odd perfect numbers likely needs no introduction.
Let $\sigma(x)=\sigma_1(x)$ denote the classical sum of divisors of the positive integer $x$, and let $I(x)=\sigma(x)/x$ denote the ...
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2
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1
answer
119
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On an inequality involving the abundancy index of the Eulerian component $p^k$ of an odd perfect number $p^k m^2$
The topic of odd perfect numbers likely needs no introduction.
Denote the classical sum of divisors of the positive integer $x$ by $\sigma(x)=\sigma_1(x)$, the abundancy index of $x$ by $I(x)=\sigma(x)...
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2
votes
1
answer
137
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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)...
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232
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On odd perfect numbers $p^k m^2$ with special prime $p$, satisfying $m^2 - p^k = 2^r t$
In what follows, we will denote the classical sum of divisors of the positive integer $x$ by $\sigma(x)=\sigma_1(x)$. Denote the abundancy index of $x$ by $I(x)=\sigma(x)/x$. A positive integer $y$ ...
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2
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123
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If $p^k m^2$ is an odd perfect number, then is there a constant $D$ such that $\frac{\sigma(m^2)}{p^k} > \frac{m^2 - p^k}{D}$?
(Note: This question is an offshoot of this closely related one.)
Denote the classical sum of divisors of the positive integer $x$ by $\sigma(x)=\sigma_1(x)$.
The topic of odd perfect numbers likely ...
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2
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1
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66
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Follow-up to question 3121498, asked in February 2019
Let $n = p^k m^2$ be an odd perfect number given in Eulerian form (i.e. $p$ is the special/Euler prime satisfying $p \equiv k \equiv 1 \pmod 4$ and $\gcd(p,m)=1$). Denote the classical sum of ...
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1
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1
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70
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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 $q$
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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2
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3
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128
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Applying a criterion on deficient numbers to the proper factors of an odd perfect number
Hereinafter, 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 $x$. Denote the deficiency of $x$ by
$$D(...
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