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**Arnie Dris's Publications - 3rd Quarter, 2018**

Conditions equivalent to the Descartes–Frenicle–Sorli Conjecture on odd perfect numbers – Part II (co-authored with Doli-Jane Uvales Tejada)

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**On the abundancy index/outlaw status of the fraction $\frac{p+2}{p}$, where $p$ is an odd prime**

As before, we consider the equation $$\frac{\sigma(x)}{x} = \frac{p+2}{p}$$ where $p$ is an odd prime. We consider several cases. (Note that the list of cases presented here is not exhaustive.) Case 1: $3 \mid x$ Since $$\frac{\sigma(x)}{x} = \frac{p+2}{...

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**Arnie Dris's Publications - 2nd Quarter, 2018**

The Non-Euler Part of a Spoof Odd Perfect Number is not Almost Perfect (co-authored with Doli-Jane T. Lugatiman)

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**Would like to get numerical (lower [and upper?]) bounds for $p$**

(This post is copied verbatim from this MSE question .) This question is an offshoot of this earlier MSE question . Let $\sigma(z)$ denote the sum of divisors of $z \in \mathbb{N}$, the set of positive integers. Denote the abundancy index of $z$ by $I(z) :...

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**Can the following argument be pushed to a full proof that $(p + 2)/p$ is an outlaw if $p$ is an odd prime?**

(The following post is extracted verbatim from this MSE question .) This is related to this earlier MSE question . Let $\sigma(x)$ be the sum of the divisors of $x$, and denote the abundancy index of $x$ by $I(x):=\sigma(x)/x$. If the equation $I(a) = b/c$ ...

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**If $q^k n^2$ is an odd perfect number with Euler prime $q$, does this equation imply that $k=1$?**

(Note: This post was copied verbatim from this MSE question .) Let $\sigma(x)$ be the sum of the divisors of $x$. Denote the deficiency of $x$ by $D(x) := 2x - \sigma(x)$, and the sum of the aliquot divisors of $x$ by $s(x) := \sigma(x) - x$. Here is my q...

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**If $\frac{σ(x)}{x}=\frac{p+2}p$ where $p$ is an odd prime, does it follow that $x$ is an odd square?**

(Note: The following proof was copied verbatim from the answer of MSE user Alex Francisco .) First, note that for any coprime $a, b \in \mathbb{N}_+$, there is$$ I(ab) = I(a) I(b). $$ Suppose there is an even number $n = 2^k \cdot l$, where $k \geq 1$ and ...

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**Arnie Dris's Publications - 3rd Quarter, 2017**

The abundancy index of divisors of odd perfect numbers – Part III The Abundancy Index of Divisors of Spoof Odd Perfect Numbers

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**OPN Research - October 2017**

If there are infinitely many odd perfect numbers $q^k n^2$ with Euler prime $q$, then by a contrapositive to Dris and Luca's result (Dris, Luca 2017) , it follows that $\sigma(n^2)/q^k < K$ does not hold for any $K \in \mathbb{N}$. This implies that $$\fra...

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**$|\text{Odd Perfect Numbers }| < \infty \Longrightarrow |\text{Even Perfect Numbers}| < \infty$**

Let $q^k n^2$ be an odd perfect number given in Eulerian form, and let $(2^p - 1){2^{p-1}}$ be an even perfect number given in Euclidean form. Denote the abundancy index of $x \in \mathbb{N}$ as $I(x)=\sigma(x)/x$, where $\sigma(x)$ is the sum of the divis...

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