Can this bound for the abundancy index of $n$ be improved, given that $q^k n^2$ is an odd perfect number with $k=1$?

Question

In what follows, set $I(x)=\sigma(x)/x$ to be the abundancy index of $x \in \mathbb{N}$, where $\sigma(x)$ is the sum of divisors of $x$. If $I(y)=2$ and $y$ is odd, then $y$ is called an odd perfect number. Euler proved that an odd perfect number, if one exists, must have the so-called Eulerian form $N = q^k n^2$ where $q$ is the special/Euler prime satisfying $q \equiv k \equiv 1 \pmod 4$ and $\gcd(q,n)=1$.

The following result was communicated to me (via e-mail, by Pascal Ochem) on April 17, 2013.

If $N = {q^k}{n^2}$ is an odd perfect number given in Eulerian form, then $$I(n) > {\left(\frac{8}{5}\right)}^{\frac{\ln(4/3)}{\ln(13/9)}} \approx 1.44440557.$$

The proof of this result uses the following lemma.

Let $x(n) = \ln(I(n^2))/\ln(I(n))$. If $\gcd(a, b) = 1$, then $$\min(x(a), x(b)) < x(ab) < \max(x(a),x(b)).$$

Proof of Lemma: First, note that $x(a) \neq x(b)$ (since $\gcd(a , b) = 1$). Without loss of generality, we may assume that $x(a) < x(b)$. Thus, since $$x(n) = \frac{\ln(I(n^2))}{\ln(I(n))},$$ we have $$\frac{\ln(I(a^2))}{\ln(I(a))} < \frac{\ln(I(b^2))}{\ln(I(b))}.$$ This implies that $$\ln(I(a^2))\ln(I(b)) < \ln(I(b^2))\ln(I(a)).$$ Adding $\ln(I(a^2))\ln(I(a))$ to both sides of the last inequality, we get $$\ln(I(a^2))\bigg(\ln(I(b)) + \ln(I(a))\bigg) < \ln(I(a))\bigg(\ln(I(b^2)) + \ln(I(a^2))\bigg).$$ Using the identity $\ln(X) + \ln(Y) = \ln(XY)$, we can rewrite the last inequality as $$\ln(I(a^2))\ln(I(ab)) < \ln(I(a))\ln(I((ab)^2))$$ since $I(x)$ is a weakly multiplicative function of $x$ and $\gcd(a, b) = 1$. It follows that $$\min(x(a), x(b)) = x(a) = \frac{\ln(I(a^2))}{\ln(I(a))} < \frac{\ln(I((ab)^2))}{\ln(I(ab))} = x(ab).$$

Under the same assumption $x(a) < x(b)$, we can show that $$x(ab) < x(b) = \max(x(a),x(b))$$ by adding $\ln(I(b^2))\ln(I(b))$ ( instead of $\ln(I(a^2))\ln(I(a))$ ) to both sides of the inequality $$\ln(I(a^2))\ln(I(b)) < \ln(I(b^2))\ln(I(a)).$$

This finishes the proof.

From this lemma, we note that $1 < x(n) < 2$ follows from $$I(n) < I(n^2) < {\left(I(n)\right)}^2$$ and $I(n^2) = {\left(I(n)\right)}^{x(n)}$.

<p>The trivial lower bound for <span class="math-container">$I(n)$</span> is
<span class="math-container">$${\bigg(\frac{8}{5}\bigg)}^{1/2} &lt; I(n).$$</span></p>

<p>Note that decreasing the denominator in the exponent gives an increase in the lower bound for <span class="math-container">$I(n)$</span>.</p>

Proof of Result: We sketch a proof for the mentioned result here, as communicated to the author by Pascal Ochem.

Suppose that $N = {q^k}{n^2}$ is an odd perfect number given in Eulerian form.

We want to obtain a lower bound on $I(n)$. We know that $$I(n^2) = 2/I(q^k) > 2/(5/4) = 8/5.$$ We need to improve the trivial bound $I(n^2) < {\left(I(n)\right)}^2$.

Let $x(n)$ be such that $$I(n^2) = {\bigg(I(n)\bigg)}^{x(n)}.$$

That is, $x(n) = \ln(I(n^2))/\ln(I(n))$. We want an upper bound on $x(n)$ for $n$ odd. By the lemma, we consider the component $r^s$ with $r$ prime that maximizes $x(r^s)$.

We have $$I(r^s) = \frac{r^{s + 1} - 1}{{r^s}(r - 1)} = 1 + \frac{1}{r - 1} - \frac{1}{{r^s}(r - 1)}.$$ Also, $$I(r^{2s}) = \frac{r^{2s + 1} - 1}{{r^{2s}}(r - 1)} = I(r^s)\bigg(1 + \left(\frac{1 - r^{-s}}{r^{s + 1} - 1}\right)\bigg).$$ So, $$x(r^s) = \frac{\ln(I(r^{2s}))}{\ln(I(r^s))} = \frac{\ln(I(r^s)) + \ln(1 + \left(\frac{1 - r^{-s}}{r^{s + 1} - 1}\right))}{\ln(I(r^s))},$$ from which it follows that $$x(r^s) = 1 + \frac{\ln(1 + \left(\frac{1 - r^{-s}}{r^{s + 1} - 1}\right))}{\ln(1 + \frac{1}{r - 1} - \frac{1}{{r^s}(r - 1)})}.$$ We can check that $$x(r^s) > x(r^t)$$ if $s < t$ and $r \geq 3$. Therefore, $x(r^s)$ is maximized for $s = 1$. Now, $$x(r) = 1 + \frac{\ln(1 + (1/(r(r + 1))))}{\ln(1 + (1/r))} = \frac{\ln(1 + (1/r) + (1/r)^2)}{\ln(1 + (1/r))} = \ln(I(r^2))/\ln(I(r)),$$ which is maximized for $r = 3$. So, $$x(3) = \ln(I(3^2))/\ln(I(3)) = \ln(13/9)/\ln(4/3) \approx 1.27823.$$

The claimed result then follows, and the proof is complete.

Motivation for My Inquiry

Note that, since $q$ is prime with $q \equiv 1 \pmod 4$, then $q \geq 5$. In particular, when $k=1$, then we obtain $$I(q^k) = 1 + \frac{1}{q} \leq \frac{6}{5} \iff I(n^2) = \frac{2}{I(q^k)} \geq \frac{5}{3},$$ from which we get $$I(n) > \bigg(I(n^2)\bigg)^{1/{x(3)}} \geq \bigg(\frac{5}{3}\bigg)^{\frac{\ln(4/3)}{\ln(13/9)}} \approx 1.4912789897463723558.$$

Here is my question:

Would it be possible to tweak the argument in this post in order to come up with a proof for the implication $$k=1 \implies I(n) > \frac{3}{2}?$$

Answer 1

(Not a complete answer.)

Let $q^k n^2$ be an odd perfect number with special/Euler prime $q$ satisfying $q \equiv k \equiv 1 \pmod 4$ and $\gcd(q,n)=1$.

First off, we show the following lemma.

Lemma 1 $I(n) \neq 3/2 = I(2)$

Proof The proof is trivial and follows from the fact that $n \neq 2$, since $n$ must be odd. (In fact, we know by Nielsen's latest result that $\omega(n) \geq 9$, where $\omega(n)$ is the number of distinct prime factors of $n$. In particular, this means that $n$ is composite.) Note that $2$ is solitary as it is prime.

Next, we prove the following implication.

Theorem $\bigg((k=1) \land (I(n) < 3/2)\bigg) \implies q=5$

Proof Since $k=1$, then we obtain $$\frac{2q}{q+1}=I(n^2) \leq (I(n))^{\frac{\ln(13/9)}{\ln(4/3)}} < \bigg(\frac{3}{2}\bigg)^{\frac{\ln(13/9)}{\ln(4/3)}}.$$ Using WolframAlpha, we get that $-1 < q < 5.23316$, from which we conclude that $q=5$, since $q$ is a prime satisfying $q \equiv 1 \pmod 4$.

In general, we can remove the assumption that $k=1$:

Theorem $I(n) < 3/2 \implies q=5$

Proof We proceed as follows: $$\frac{2(q-1)}{q} < I(n^2) \leq (I(n))^{\frac{\ln(13/9)}{\ln(4/3)}} < \bigg(\frac{3}{2}\bigg)^{\frac{\ln(13/9)}{\ln(4/3)}}.$$

Using WolframAlpha, we obtain $0 < q < 6.23316$, which implies that $q=5$, since $q$ is a prime number satisfying $q \equiv 1 \pmod 4$.

Remarks: Note that $I(n) > 3/2$ would also follow from the inequality $D(n) < s(n)$, where $D(n)=2n-\sigma(n)$ is the deficiency of $n$ and $s(n)=\sigma(n) - n$ is the sum of the aliquot divisors of $n$.

Answer 2

Let $N = q^k n^2$ be a hypothetical odd perfect number given in Eulerian form.

This answer attempts to prove that the inequality $I(n) > 3/2$ holds unconditionally, using a variation on the factor chain method.

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Assume to the contrary that $I(n) \leq 3/2$. As shown in the other answer, $I(n) \neq 3/2$, and that $I(n) < 3/2$ implies $q = 5$.

We get the series of implications: $$I(n) < \frac{3}{2} \implies q = 5 \implies 3 = \frac{q + 1}{2} \mid \frac{\sigma(q^k)}{2} \mid n^2 \implies 3 \mid n \text{ (since } 3 \text{ is squarefree) }$$ $$\implies \frac{4}{3} = I(3) \leq I(n) < \frac{3}{2} \text{ (no contradiction) }$$ $$3 \mid n^2 \implies 3^2 \mid n^2 \implies \sigma(3^2) \mid \sigma(n^2) \mid N = 5^k n^2$$ $$\implies 13 \mid n^2 \implies 13 \mid n \text{ (since } 13 \text{ is squarefree) }$$ $$\implies \frac{4}{3}\cdot\frac{14}{13} = I(3)I(13) \leq I(n) < \frac{3}{2} \text{ (no contradiction) }$$ $$13 \mid n^2 \implies 13^2 \mid n^2 \implies \sigma(13^2) \mid \sigma(n^2) \mid N = 5^k n^2 \implies 183 = 3\cdot{61} \mid n^2$$ $$\implies 61 \mid n^2 \implies 61 \mid n \text{ (since } 61 \text{ is squarefree) }$$ $$\implies \frac{4}{3}\cdot\frac{14}{13}\cdot\frac{62}{61} = I(3)I(13)I(61) \leq I(n) < \frac{3}{2} \text{ (no contradiction) }$$ $$61 \mid n^2 \implies 61^2 \mid n^2 \implies \sigma(61^2) \mid \sigma(n^2) \mid N = 5^k n^2 \implies 3783 = 3\cdot{13}\cdot{97} \mid n^2$$ $$\implies 97 \mid n^2 \implies 97 \mid n \text{ (since } 97 \text{ is squarefree) }$$ $$\implies \frac{4}{3}\cdot\frac{14}{13}\cdot\frac{62}{61}\cdot\frac{98}{97} = I(3)I(13)I(61)I(97) \leq I(n) < \frac{3}{2} \text{ (no contradiction) }$$ $$97 \mid n^2 \implies 97^2 \mid n^2 \implies \sigma(97^2) \mid \sigma(n^2) \mid N = 5^k n^2 \implies 9507 = 3\cdot{3169} \mid n^2$$ $$\implies 3169 \mid n^2 \implies 3169 \mid n \text{ (since } 3169 \text{ is squarefree) }$$ $$\implies I(3)I(13)I(61)I(97)I(3169) \leq I(n) < \frac{3}{2} \text{ (no contradiction) }$$ $$3169 \mid n^2 \implies 3169^2 \mid n^2 \implies \sigma(3169^2) \mid \sigma(n^2) \mid N = 5^k n^2$$ $$\implies 10045731 = 3\cdot{3348577} \mid n^2$$ $$\implies 3348577 \mid n^2 \implies 3348577 \mid n \text{ (since } 3348577 \text{ is squarefree) }$$ $$\implies I(3)I(13)I(61)I(97)I(3169)I(3348577) \leq I(n) < \frac{3}{2} \text{ (no contradiction) }$$ $$3348577 \mid n^2 \implies 3348577^2 \mid n^2 \implies \sigma(3348577^2) \mid \sigma(n^2) \mid N = 5^k n^2$$ $$\implies 11212971273507 = 3\cdot{3737657091169} \mid n^2$$ $$\implies 3737657091169 \mid n^2 \implies 3737657091169 \mid n \text{ (since } 3737657091169 \text{ is squarefree) }$$ $$\implies I(3)I(13)I(61)I(97)I(3169)I(3348577)I(3737657091169) \leq I(n) < \frac{3}{2} \text{ (no contradiction) }$$ $$3737657091169 \mid n^2 \implies 3737657091169^2 \mid n^2 \implies \sigma(3737657091169^2) \mid \sigma(n^2) \mid N = 5^k n^2$$ $$\implies 13970080531169648034877731 = 3\cdot{181}\cdot{26042690887}\cdot{987900542491} \mid n^2$$ $$\implies 181 \mid n^2 \implies 181 \mid n \text{ (since } 181 \text{ is squarefree) }$$ $$\implies 26042690887 \mid n^2 \implies 26042690887 \mid n \text{ (since } 26042690887 \text{ is squarefree) }$$ $$\implies 987900542491 \mid n^2 \implies 987900542491 \mid n \text{ (since } 987900542491 \text{ is squarefree) }$$ $$\implies I(3)I(13)I(61)I(97)I(3169)I(3348577)I(3737657091169)I(181)I(26042690887)I(987900542491) \leq I(n) < \frac{3}{2} \text{ (no contradiction) }$$ $$181 \mid n^2 \implies 181^2 \mid n^2 \implies \sigma(181^2) \mid \sigma(n^2) \mid N = 5^k n^2 \implies 32943 = 3\cdot{79}\cdot{139} \mid n^2$$ $$\implies 79 \mid n^2 \implies 79 \mid n \text{ (since } 79 \text{ is squarefree) }$$ $$\implies 139 \mid n^2 \implies 139 \mid n \text{ (since } 139 \text{ is squarefree) }$$ But this implies that $$I(3)I(13)I(61)I(97)I(3169)I(3348577)I(3737657091169)I(181)I(26042690887)I(987900542491)I(79)I(139) \leq I(n) < \frac{3}{2},$$ where we have the approximation $$I(3)I(13)I(61)I(97)I(3169)I(3348577)I(3737657091169)I(181)I(26042690887)I(987900542491)I(79)I(139) \approx 1.512675290866383731626612,$$ which is a contradiction.

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We therefore conclude that the inequality $$I(n) > \frac{3}{2}$$ must hold unconditionally.