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Tagged with divisor-sum gcd-and-lcm
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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 ...
- 8,010
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Upper bounds on the greatest common divisor of sums of geometric series
Let $S_1=\sum_{i=0}^{n} p^i = \frac{p^{n+1}-1}{p-1}$ and $S_2=\sum_{i=0}^{m} q^i = \frac{q^{n+1}-1}{q-1}$ be two sums of geometric series, and $\gcd\left(S_1,S_2\right)$ its greatest common divisor. ...
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Proving $n \mid \sigma(n^2)$
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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If $p^k m^2$ is an odd perfect number, then $D(p^k)/s(p^k)$ is in lowest terms. Does this contradict $D(p^k)D(m^2)=2s(p^k)s(m^2)$?
In what follows, denote the classical sum of divisors of the positive integer $x$ by $\sigma(x)=\sigma_1(x)$.
If $n$ is odd and $\sigma(n)=2n$, then $n$ is called an odd perfect number.
Euler showed ...
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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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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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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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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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Particular values for the sum of divisors function from billiards
In this post we consider as reference the article Arithmetic billiards from Wikipedia. We consider the arithmetic billiard that is explained in the article.
I've wondered if we can compute some simple ...
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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 ...
- 8,010
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On odd perfect numbers and a GCD - Part VII
(Pardon me for being somewhat stubborn, but this question will be the last for this week. This post is an offshoot of this one.)
Let $N = q^k n^2$ be an odd perfect number be an odd perfect number ...
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Does $G \mid I$ and $I \mid H$ still hold if $\sigma(q^k)/2$ is not squarefree, where $q^k n^2$ is an odd perfect number with special prime $q$?
This question is an offshoot of this post #1 and this post #2.
Let $N = q^k n^2$ be an odd perfect number be an odd perfect number with special prime $q$ satisfying $q \equiv k \equiv 1 \pmod 4$ and $\...
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On odd perfect numbers and a GCD - Part VI
(Note: This post is closely related to this earlier MSE question.)
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 $\gcd(q,n)=1$....
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