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I have no idea of chemistry, just a little bit what I know from school, but I found through some search some connection between entropy (Shannon entropy) of the divisors of some natural numbers and the Madelung constants.

My question is this:

Are these numbers Madelung constants?

1 -0.000000000000000
2 1.38629436111989 ( http://oeis.org/A016627 )
3 1.38629436111989
4 3.31573052232910 
5 1.38629436111989
6 5.54531864369184  ( http://oeis.org/A257872 )
7 1.38629436111989
8 5.57849454296371
9 3.29975689063759
10 5.54521210457048
11 1.38629436111989
12 10.7943529563000
13 1.38629436111989
14 5.54518969515687
15 5.54519368543309
16 8.08924328274917
17 1.38629436111989
18 10.7599543952466
19 1.38629436111989
20 10.7921021686465

Thanks for your help!

Related question: https://mathoverflow.net/questions/346239/entropy-magnitude-diversity-of-finite-metric-spaces-in-number-theory

Edit: I think I found an explanation for the numbers ( entropy ). It is approximately:

$\tau(n) \log(\tau(n))$, where $\tau(n)$ is the number of divisors of $n$.

So the $1.386$ and $5.545$ are just coincidences with Madelung constants I think.

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  • $\begingroup$ 1.386 is 2log2 but what about 5.545 the closet approximation is 7log7 which is still 5.91. Am i missing something? $\endgroup$ – user78585 Nov 18 at 8:53
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    $\begingroup$ $5.545=8 \log(2)$ see the oeis link which relates this number to one Madelung contant $\endgroup$ – orgesleka Nov 18 at 9:01
  • $\begingroup$ Ok i was mistaking that both the numbers inside and outside the log must be the same $\endgroup$ – user78585 Nov 18 at 9:30
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I don't know if this would be very helpful or not but as i had some free time on hand I got a list of Madelung Constant Values from the Internet

In the list most of values were reported to 3-4 decimal places so I have Compared only the first decimal places as once the second decimal place was used none of the values matched. Maybe this would be a helpful start.

$$M_a$$ of ZnS (Wurtzite Hexagonal) is $10.7153$ (nearby the 12th and 18th entry in your table)

As Oscar Lanzi has commented $1.386$ is approximately $$2\log(2)$$ which is the Madelung constant for a one-dimensional line of alternating charges

As the question has been edited to add further details maybe the second one is a coincidence.

Only Nearby Matches So Far

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    $\begingroup$ thanks for your answer $\endgroup$ – orgesleka Nov 17 at 18:00
  • $\begingroup$ @orgesleka You're Welcome $\endgroup$ – user78585 Nov 17 at 18:04
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    $\begingroup$ The $1.386$ figure is $2\log 2$, which is the Madelung constant for a one-dimensional line of alternating charges. $\endgroup$ – Oscar Lanzi Nov 17 at 19:52
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    $\begingroup$ I say add it in. $\endgroup$ – Oscar Lanzi Nov 17 at 20:01
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    $\begingroup$ @OscarLanzi Done $\endgroup$ – user78585 Nov 17 at 20:25

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