All Questions
Tagged with divisor-sum arithmetic-functions
153
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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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$\sum_{k = 1}^{\infty} k\lfloor\frac{n}{k} \rfloor = 1 + \sum_{k = 1}^n \sigma_1(n)$
For any $f: \Bbb{N} \to \Bbb{Z}$ there exists a unique transformed function $F:\Bbb{N} \to \Bbb{Z}$ such that:
$$
f(n) = \sum_{k = 1}^{\infty}F_k\lfloor\frac{n}{k}\rfloor
$$
For example, set $F_1 = f(...
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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 proof for the divisibility constraint $\sigma(q^k)/2 \mid n$ correct, 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 given in Eulerian form, i.e. $q$ is the special prime satisfying $q \equiv k \equiv 1 \pmod 4$ and $\gcd(q,n)=1$.
Define the GCDs
$$G = \gcd(\sigma(q^k),\...
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Is this disproof for the Descartes-Frenicle-Sorli Conjecture that $k=1$, if $p^k m^2$ is an odd perfect number, valid?
Let $N = p^k m^2$ be an odd perfect number with special prime $p$ satisfying $p \equiv k \equiv 1 \pmod 4$ and $\gcd(p,m)=1$.
It is known that
$$D(p^k)D(m^2)=2s(p^k)s(m^2) \tag{0}$$
where $D(x)=2x-\...
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Is there an analytical solution to the inequality $\frac{1 + k({p}^{(k+1)/2)})}{p^k} < \frac{p}{p - 1}$ if one were to bound $k$ in terms of $p$?
My question is as is in the title:
Is there an analytical solution to the inequality
$$\frac{1 + k({p}^{(k+1)/2)})}{p^k} < \frac{p}{p - 1}$$ if one were to bound $k$ in terms of $p$?
Here, $p \...
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Why is it that, if there are no odd perfect numbers, then there are no other $3$-perfect numbers, apart from the six known, as of the year $1643$?
Let $\sigma(x)=\sigma_1(x)$ denote the classical sum of divisors of the positive integer $x$.
A number $N$ is said to be $k$-perfect if $\sigma(N)=kN$ where $k$ is a positive integer.
The number $1$ ...
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On the prime factorization of $n$ and the quantity $J = \frac{n}{\gcd(n,\sigma(q^k)/2)}$, where $q^k n^2$ is an odd perfect number
Let $N = q^k n^2$ be an odd perfect number with special prime $q$ satisfying $q \equiv k \equiv 1 \pmod 4$ and $\gcd(q,n)=1$. Denote the classical sum of divisors of the positive integer $x$ by $\...
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On a consequence of $G \mid I \iff \gcd(G, I) = G$ (Re: Odd Perfect Numbers and GCDs)
Let $N = q^k n^2$ be an odd perfect number given in the so-called Eulerian form, where $q$ is the special prime satisfying $q \equiv k \equiv 1 \pmod 4$ and $\gcd(q,n)=1$. Denote the classical sum of ...
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Can this inequality involving the deficiency and sum of aliquot divisors be improved? - Part II
This MSE question (from April 2020) asked whether the inequality
$$\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$$
where $q$ is the special prime satisfying $q \equiv k \equiv 1 \pmod 4$ and $\gcd(q,n)=1$.
In what follows, let ...
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If $2^r b^2$ is an even almost perfect number that is NOT a power of two, does it follow that $r=1$?
(The following are taken from this preprint by Antalan and Dris.)
Antalan and Tagle showed that an even almost perfect number $n \neq 2^t$ must necessarily have the form $2^r b^2$ where $r \geq 1$, $\...
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Does $\sigma(n^2)/q \mid q^k n^2$ imply $\sigma(n^2)/q \mid n^2$, if $q^k n^2$ is an odd perfect number with special prime $q$? - Part II
(Preamble: This inquiry is an offshoot of this MSE question.)
Denote the classical sum of divisors of the positive integer $x$ by $\sigma(x)=\sigma_1(x)$. Denote the abundancy index of $x$ as $I(x)=\...
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Are these valid proofs for the equation $\gcd(n,\sigma(n^2))=\gcd(n^2,\sigma(n^2))$, if $q^k n^2$ is an odd perfect number with special prime $q$?
Let $N = q^k n^2$ be an odd perfect number with special prime $q$ satisfying $q \equiv k \equiv 1 \pmod 4$.
It is known that
$$i(q)=\gcd(n^2,\sigma(n^2))=\frac{\sigma(n^2)}{q^k}=\frac{n^2}{\sigma(q^k)/...
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