0
$\begingroup$

I currently do not have enough computing power, so please pardon me for my question, which occurred just recently to me.

So here it goes:

Is the Descartes spoof $$\mathscr{D} = {3^2}\cdot{7^2}\cdot{{11}^2}\cdot{{13}^2}\cdot{22021} = 198585576189$$ a member of OEIS sequence A228059?

There is an existing Mathematica code in the OEIS hyperlink to test this. Thanks!

$\endgroup$
  • $\begingroup$ Note that $22021 = {{19}^2}\cdot{61}$, $\endgroup$ – Jose Arnaldo Bebita-Dris Aug 3 '18 at 6:56
  • 1
    $\begingroup$ Looking at the Mathematica code, I suppose that it will take a long time to generate the 10th, 11th, ... terms. $\endgroup$ – Claude Leibovici Aug 3 '18 at 7:57
  • $\begingroup$ Yes, essentially that is the problem, @ClaudeLeibovici. Note that the Descartes spoof might be the 10th or 11th term. Who knows? =) $\endgroup$ – Jose Arnaldo Bebita-Dris Aug 3 '18 at 8:14
  • 1
    $\begingroup$ As you say, who knows ? $\endgroup$ – Claude Leibovici Aug 3 '18 at 8:38
  • $\begingroup$ I have just also posted a closely related question here. $\endgroup$ – Jose Arnaldo Bebita-Dris Aug 3 '18 at 9:05
1
$\begingroup$

The answer to my question is NO, since the abundancy index $I(x):=\sigma(x)/x$ (where $\sigma(x)$ is the sum of the divisors of $x \in \mathbb{N}$) of the first $9$ terms of OEIS sequence A228059 are:

$$I(45) = \frac{26}{15} \approx 1.73333$$ WolframAlpha computation here $$I(405) = \frac{242}{135} \approx 1.79259$$ WolframAlpha computation here $$I(2205) = \frac{494}{245} \approx 2.01633$$ WolframAlpha computation here $$I(26325) = \frac{52514}{26325} \approx 1.99483$$ WolframAlpha computation here $$I(236925) = \frac{474362}{236925} \approx 2.00216$$ WolframAlpha computation here $$I(1380825) = \frac{307086}{153425} \approx 2.00154$$ WolframAlpha computation here $$I(1660725) = \frac{3323138}{1660725} \approx 2.00102$$ WolframAlpha computation here $$I(35698725) = \frac{71396534}{35698725} \approx 1.99997$$ WolframAlpha computation here $$I(3138290325) = \frac{77488034}{38744325} \approx 1.99998$$ WolframAlpha computation here

Notice that, by the definition of OEIS sequence A228059, $|I(x_i)-2|$ must be a (strictly?) decreasing sequence.

Therefore, since $$I(198585576189) = \frac{23622}{11011} \approx 2.14531,$$ it follows that the Descartes spoof $$\mathscr{D} = 198585576189$$ is not a member of OEIS sequence A228059.

Added August 15 2018

In an e-mail correspondence, Tony D. Noe (author of OEIS sequence A228059) says that "(he) found the next term for this sequence: $29891138805 = {5}\cdot({3^2}\cdot{{11}^2}\cdot{71})^2$, (and that) (i)t took $5$ days on a fairly fast Mac."

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.