APOD: From Orion to the Southern Cross (2022 Jan 18)

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APOD: From Orion to the Southern Cross (2022 Jan 18)

Post by APOD Robot » Tue Jan 18, 2022 5:05 am

Image From Orion to the Southern Cross

Explanation: This is a sky filled with glowing icons. On the far left is the familiar constellation of Orion, divided by its iconic three-aligned belt stars and featuring the famous Orion Nebula, both partly encircled by Barnard's Loop. Just left of center in the featured image is the brightest star in the night: Sirius. Arching across the image center is the central band of our Milky Way Galaxy. On the far right, near the top, are the two brightest satellite galaxies of the Milky Way: the Large Magellanic Cloud (LMC), and the Small Magellanic Cloud (SMC). Also on the far right -- just above the cloudy horizon -- is the constellation of Crux, complete with the four stars that make the iconic Southern Cross. The featured image is a composite of 18 consecutive exposures taken by the same camera and from the same location in eastern Australia during the last days of last year. In the foreground, picturesque basalt columns of the Bombo Quarry part to reveal the vast Pacific Ocean.

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Re: APOD: From Orion to the Southern Cross (2022 Jan 18)

Post by jks » Tue Jan 18, 2022 5:21 am

Nice detail when zoomed in. I wonder what the streak of light is that is to the left of what I think is Canopus, the brightest unlabeled star.

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Re: APOD: From Orion to the Southern Cross (2022 Jan 18)

Post by Ann » Tue Jan 18, 2022 5:41 am

jks wrote: Tue Jan 18, 2022 5:21 am Nice detail when zoomed in. I wonder what the streak of light is that is to the left of what I think is Canopus, the brightest unlabeled star.
That is indeed Canopus. The streak? Maybe a meteor.

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Re: APOD: From Orion to the Southern Cross (2022 Jan 18)

Post by Ann » Tue Jan 18, 2022 6:16 am

I really like today's APOD, and the full size version is great. But me being me, I accidentally discovered something about it.

One of my absolutely favorite clusters is NGC 2362 in Canis Major, and I decided to find it in the full size version of the APOD.

NGC 2362 and VdB 96 Velimir Popov and Emil Ivanov 2016.png
NGC 2362 and friends. Velimir Popov and Emil Ivanov.
Detail from APOD January 18 2022 NGC 2362.png
Epsilon, Sigma and Delta CMa - and NGC 2362, DOUBLED?

So, yes. I think NGC 2362 has been doubled in today's APOD. I'm sure that the "true" NGC 2362 is located just below where I have written NGC 2362. To the left of NGC 2362 is 29 CMa. Please note that 29 CMa is located to the right of NGC 2362 in the Popov/Ivanov image, but to the left of NGC 2362 in the APOD.

But look at the blue object to the right of NGC 2362 in the APOD. I checked and checked my software, and there is, indeed, no bright blue object near NGC 2362 "on the opposite side of 29 CMa". There shouldn't be anything there, but in the APOD, there is indeed something there. To me that something looks exactly like NGC 2362, doubled.

Sorry for the nitpick, but... I can't help myself, I couldn't resist looking for NGC 2362! It's lovely, isn't it?


Ann
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Re: APOD: From Orion to the Southern Cross (2022 Jan 18)

Post by Chris Peterson » Tue Jan 18, 2022 9:15 am

Ann wrote: Tue Jan 18, 2022 6:16 am I really like today's APOD, and the full size version is great. But me being me, I accidentally discovered something about it.

One of my absolutely favorite clusters is NGC 2362 in Canis Major, and I decided to find it in the full size version of the APOD.

So, yes. I think NGC 2362 has been doubled in today's APOD.
It is almost impossible to avoid compositing errors in widefield mosaics like this. Even with modern tools like PixInsight that largely solve for image distortion and mostly "undo" it, some mismatch along boundaries is pretty unavoidable. And if you happen to end up with an identifiable object along one of those boundaries instead of relatively uniform star fields... well.
Chris

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Re: APOD: From Orion to the Southern Cross (2022 Jan 18)

Post by particles@mweb.co.za » Tue Jan 18, 2022 12:21 pm

My Submission bounced.

Here its is again - maybe a little different.

I said that in your description you missed the Great Southern Triangle - Sirius, Procyon and Betelgeuse - in the southern hemisphere this surpasses the Great Square of Pegasus

Many thanks for APOD - great

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Re: APOD: From Orion to the Southern Cross (2022 Jan 18)

Post by orin stepanek » Tue Jan 18, 2022 12:45 pm

OrionCross_LucyH_1080_annotated.jpg
I loved it! fantastic photo and I loved the annotations explaining the
items in the Milky Way! Kudos to the photographer!:D
bigstock-Young-Crazy-Surprised-Cat-Make-233591413.jpg
kitty likes it too! 🤩
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Re: APOD: From Orion to the Southern Cross (2022 Jan 18)

Post by Will » Tue Jan 18, 2022 2:36 pm

Is the image inverted? It seems like directions are reversed compared to what I see in the sky chart at Heavens Above for a viewing location of Bombo. For example, the sky chart shows (and my expectation is) that Sirius is to the left of Orion, but Sirius is to the right of Orion in the image. What am I doing wrong?

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Re: APOD: From Orion to the Southern Cross (2022 Jan 18)

Post by Chris Peterson » Tue Jan 18, 2022 2:49 pm

Will wrote: Tue Jan 18, 2022 2:36 pm Is the image inverted? It seems like directions are reversed compared to what I see in the sky chart at Heavens Above for a viewing location of Bombo. For example, the sky chart shows (and my expectation is) that Sirius is to the left of Orion, but Sirius is to the right of Orion in the image. What am I doing wrong?
"Left" and "right" are nebulous concepts on sky maps. They can change with time. Sirius is east of Orion, and that's what we see in the image. This looks about right for a view to the southeast in early evening at the end of December.
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Re: APOD: From Orion to the Southern Cross (2022 Jan 18)

Post by neufer » Tue Jan 18, 2022 5:08 pm

https://en.wikipedia.org/wiki/Bombo_Headland_Quarry_Geological_Site wrote:
Click to play embedded YouTube video.

<<In 1979 the importance of Bombo Quarry's geological features was brought to the New South Wales Heritage Council's attention. The quarry was owned by the Metropolitan Water, Sewerage and Drainage Board and it was proposed that a pollution control plant be constructed on the floor of the disused quarry. It also made an appearance in Mighty Morphin Power Rangers: The Movie as part of the planet Phaedos:

Ten thousand years ago, Zordon clashes with his nemesis, Rita Repulsa, on Earth. During their final battle, Rita traps Zordon in a time warp while Zordon seals Rita and her minions away in a "Dumpster" on the Moon. After the battle ends, Zordon, with the aid of his robot assistant, Alpha 5, creates a California Command Center. When Rita Repulsa is released from the Dumpster, Zordon recruits five teenagers fto become the Power Rangers. Zordon's assistant Alpha 5 sends the Rangers to the distant planet Phaedos to obtain the Great Power and save Zordon. On the Moon, Ivan usurps Rita and Zedd, trapping them in a snow globe, and then sends his Tengu warriors to Phaedos and begins building an army. He uses children to bring his ooze to their parents, and it hypnotizes them into becoming his workforce to dig up his Ecto-Morphicon Titans. When Fred Kelman, a friend of the Rangers', discovers his father missing, he finds him working at the construction site. Following site inspections and lengthy consultations between the Metropolitan Water, Sewerage and Drainage Board, NSW Heritage Council and other key agencies a Permanent Conservation Order was placed over the site in 1983. It was transferred onto the State Heritage Register in 1999.>>
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Re: APOD: From Orion to the Southern Cross (2022 Jan 18)

Post by johnnydeep » Wed Jan 19, 2022 1:55 am

Chris Peterson wrote: Tue Jan 18, 2022 9:15 am
Ann wrote: Tue Jan 18, 2022 6:16 am I really like today's APOD, and the full size version is great. But me being me, I accidentally discovered something about it.

One of my absolutely favorite clusters is NGC 2362 in Canis Major, and I decided to find it in the full size version of the APOD.

So, yes. I think NGC 2362 has been doubled in today's APOD.
It is almost impossible to avoid compositing errors in widefield mosaics like this. Even with modern tools like PixInsight that largely solve for image distortion and mostly "undo" it, some mismatch along boundaries is pretty unavoidable. And if you happen to end up with an identifiable object along one of those boundaries instead of relatively uniform star fields... well.
And well spotted, Ann! I wonder what else might be duplicated.
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Re: APOD: From Orion to the Southern Cross (2022 Jan 18)

Post by neufer » Wed Jan 19, 2022 3:52 pm

Ann wrote: Tue Jan 18, 2022 6:16 am
One of my absolutely favorite clusters is NGC 2362 in Canis Major, and I decided to find it in the full size version of the APOD.

I think NGC 2362 has been doubled in today's APOD.
https://en.wikipedia.org/wiki/Amicable_numbers wrote:
  • factors of 2362 = 1, 2, and 1181
    . [1+2+1181 = 1184]

    factors of 1184 = 1, 2, 4, 8, 16, 32, 37, 74, 148, 296, and 592

    . [1+2+4+8+16+32+37+74+148+296+592 = 1210]

    factors of 1210 = 1, 2, 5, 10, 11, 22, 55, 110, 121, 242, and 605

    . [1+2+5+10+11+22+55+110+121+242+605 = 1184]
Click to play embedded YouTube video.
johnnydeep wrote: Wed Jan 19, 2022 1:55 am
And well spotted, Ann! I wonder what else might be duplicated.
Chris Peterson wrote: Tue Jan 18, 2022 9:15 am
It is almost impossible to avoid compositing errors in widefield mosaics like this. Even with modern tools like PixInsight that largely solve for image distortion and mostly "undo" it, some mismatch along boundaries is pretty unavoidable. And if you happen to end up with an identifiable object along one of those boundaries instead of relatively uniform star fields... well.
http://www.marianotomatis.it/blog.php?p ... on=english
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Re: APOD: From Orion to the Southern Cross (2022 Jan 18)

Post by johnnydeep » Wed Jan 19, 2022 5:37 pm

neufer wrote: Wed Jan 19, 2022 3:52 pm
Ann wrote: Tue Jan 18, 2022 6:16 am
One of my absolutely favorite clusters is NGC 2362 in Canis Major, and I decided to find it in the full size version of the APOD.

I think NGC 2362 has been doubled in today's APOD.
https://en.wikipedia.org/wiki/Amicable_numbers wrote:
  • factors of 2362 = 1, 2, and 1181
    . [1+2+1181 = 1184]

    factors of 1184 = 1, 2, 4, 8, 16, 32, 37, 74, 148, 296, and 592

    . [1+2+4+8+16+32+37+74+148+296+592 = 1210]

    factors of 1210 = 1, 2, 5, 10, 11, 22, 55, 110, 121, 242, and 605

    . [1+2+5+10+11+22+55+110+121+242+605 = 1184]
So, 1184 and 1210 are a true amicable pair, but I guess 2362 is "amicable once removed", or an "amicable pair generator", or "a third wheel" or something like that.
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Re: APOD: From Orion to the Southern Cross (2022 Jan 18)

Post by neufer » Wed Jan 19, 2022 9:51 pm

johnnydeep wrote: Wed Jan 19, 2022 5:37 pm
neufer wrote: Wed Jan 19, 2022 3:52 pm
Ann wrote: Tue Jan 18, 2022 6:16 am
One of my absolutely favorite clusters is NGC 2362 in Canis Major, and I decided to find it in the full size version of the APOD.

I think NGC 2362 has been doubled in today's APOD.
https://en.wikipedia.org/wiki/Amicable_numbers wrote:
  • factors of 2362 = 1, 2, and 1181
    . [1+2+1181 = 1184]

    factors of 1184 = 1, 2, 4, 8, 16, 32, 37, 74, 148, 296, and 592

    . [1+2+4+8+16+32+37+74+148+296+592 = 1210]

    factors of 1210 = 1, 2, 5, 10, 11, 22, 55, 110, 121, 242, and 605

    . [1+2+5+10+11+22+55+110+121+242+605 = 1184]
So, 1184 and 1210 are a true amicable pair, but I guess 2362 is "amicable once removed",
or an "amicable pair generator", or "a third wheel" or something like that.
It's a number whose aliquot sequence spirals into the amiable pair: {1184 <=> 1210} like:
  • 2152 => 1898 => {1210 <=> 1184}
    2122 => 1064 => 1336 => {1184 <=> 1210}
    2008 => 1772 => 1336 => {1184 <=> 1210}
    1816 => 1604 => {1210 <=> 1184}
    1690 => 1604 => {1210 <=> 1184}
    1490 => {1210 <=> 1184}
    1420 => 1604 => {1210 <=> 1184}
    1308 => 1772 => 1336 => {1184 <=> 1210}
    1064 => 1336 => {1184 <=> 1210}
[Note: There are certainly more and/or longer "arms" to the {1184 <=> 1210} "spiral galaxy"
(i.e., [1+14892] => 1490 => {1210 <=> 1184})
but they must start at numbers larger than 2362.]
https://oeis.org/A121508 wrote:
Conjectured list of numbers whose aliquot sequence eventually reaches a
sociable number cycle of length two or more, but which are not themselves part of the cycle.

562, 1064, 1188, 1308, 1336, 1380, 1420, 1490, 1604, 1690, 1692, 1772, 1816, 1898, 2008, 2122, 2152, 2172, 2362, 2542, 2630, 2652, 2676, 2678, 2856, 2930, 2950, 2974, 3124, 3162, 3202, 3278, 3286, 3332, 3350, 3360, 3596, 3712, 3750, 3850, 3938, 3944...
https://en.wikipedia.org/wiki/Sociable_number wrote:
<<In mathematics, sociable numbers are numbers whose aliquot sums form a cyclic sequence that begins and ends with the same number. They are generalizations of the concepts of amicable numbers and perfect numbers. The first two sociable sequences, or sociable chains, were discovered and named by the Belgian mathematician Paul Poulet in 1918. In a sociable sequence, each number is the sum of the proper divisors of the preceding number, i.e., the sum excludes the preceding number itself. For the sequence to be sociable, the sequence must be cyclic and return to its starting point.>>

http://factordb.com/sequences.php?se=1& ... fr=0&to=40
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Re: APOD: From Orion to the Southern Cross (2022 Jan 18)

Post by johnnydeep » Thu Jan 20, 2022 3:14 pm

neufer wrote: Wed Jan 19, 2022 9:51 pm
johnnydeep wrote: Wed Jan 19, 2022 5:37 pm
neufer wrote: Wed Jan 19, 2022 3:52 pm
So, 1184 and 1210 are a true amicable pair, but I guess 2362 is "amicable once removed",
or an "amicable pair generator", or "a third wheel" or something like that.
It's a number whose aliquot sequence spirals into the amiable pair: {1184 <=> 1210} like:
  • 2152 => 1898 => {1210 <=> 1184}
    2122 => 1064 => 1336 => {1184 <=> 1210}
    2008 => 1772 => 1336 => {1184 <=> 1210}
    1816 => 1604 => {1210 <=> 1184}
    1690 => 1604 => {1210 <=> 1184}
    1490 => {1210 <=> 1184}
    1420 => 1604 => {1210 <=> 1184}
    1308 => 1772 => 1336 => {1184 <=> 1210}
    1064 => 1336 => {1184 <=> 1210}
[Note: There are certainly more and/or longer "arms" to the {1184 <=> 1210} "spiral galaxy"
(i.e., [1+14892] => 1490 => {1210 <=> 1184})
but they must start at numbers larger than 2362.]
https://oeis.org/A121508 wrote:
Conjectured list of numbers whose aliquot sequence eventually reaches a
sociable number cycle of length two or more, but which are not themselves part of the cycle.

562, 1064, 1188, 1308, 1336, 1380, 1420, 1490, 1604, 1690, 1692, 1772, 1816, 1898, 2008, 2122, 2152, 2172, 2362, 2542, 2630, 2652, 2676, 2678, 2856, 2930, 2950, 2974, 3124, 3162, 3202, 3278, 3286, 3332, 3350, 3360, 3596, 3712, 3750, 3850, 3938, 3944...
https://en.wikipedia.org/wiki/Sociable_number wrote:
<<In mathematics, sociable numbers are numbers whose aliquot sums form a cyclic sequence that begins and ends with the same number. They are generalizations of the concepts of amicable numbers and perfect numbers. The first two sociable sequences, or sociable chains, were discovered and named by the Belgian mathematician Paul Poulet in 1918. In a sociable sequence, each number is the sum of the proper divisors of the preceding number, i.e., the sum excludes the preceding number itself. For the sequence to be sociable, the sequence must be cyclic and return to its starting point.>>

http://factordb.com/sequences.php?se=1& ... fr=0&to=40
More cool stuff. But what do you mean by "[1+14892] => 1490 => {1210 <=> 1184})"? That if you start with 1+14892 = 2,217,122 and find its factors and add them, and continue that process, that you will eventually end in the {1210 <=> 1184} amicable pair? If so, that seems hard to believe since the numbers start getting pretty big. But maybe that's what makes this even cooler...
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Re: APOD: From Orion to the Southern Cross (2022 Jan 18)

Post by neufer » Thu Jan 20, 2022 5:48 pm

johnnydeep wrote: Thu Jan 20, 2022 3:14 pm
More cool stuff. But what do you mean by "[1+14892] => 1490 => {1210 <=> 1184})"? That if you start with 1+14892 = 2,217,122 and find its factors and add them, and continue that process, that you will eventually end in the {1210 <=> 1184} amicable pair? If so, that seems hard to believe since the numbers start getting pretty big. But maybe that's what makes this even cooler...
  • 1489 is a prime number.

    The only proper factors of the square of a prime number p are: 1 & p.

    Hence, [1+p2] => [1 + p]
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Re: APOD: From Orion to the Southern Cross (2022 Jan 18)

Post by bystander » Thu Jan 20, 2022 6:49 pm

neufer wrote: Thu Jan 20, 2022 5:48 pm
  • 1489 is a prime number.

    The only proper factors of the square of a prime number p are: 1 & p.

    Hence, [1+p2] => [1 + p]

I don't think that follows. For example, let p = 13

[170] ??? [14}
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Re: APOD: From Orion to the Southern Cross (2022 Jan 18)

Post by johnnydeep » Thu Jan 20, 2022 7:44 pm

bystander wrote: Thu Jan 20, 2022 6:49 pm
neufer wrote: Thu Jan 20, 2022 5:48 pm
  • 1489 is a prime number.

    The only proper factors of the square of a prime number p are: 1 & p.

    Hence, [1+p2] => [1 + p]

I don't think that follows. For example, let p = 13

[170] ??? [14}
And more than that, for 1489 in particular, 1+1489*1489=2217122, the factors of that are {2, 1108561}, and the factors of 2+1108561 are {1, 3, 163, 489, 2267, 6801, 369521}, etc. There are some primes of the form 1+p2 but it is a hard - unsolved! - problem to prove that there are infinitely many such primes. See https://en.wikipedia.org/wiki/Landau's_problems

(Factors from https://www.mathsisfun.com/numbers/fact ... .html#calc)
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Re: APOD: From Orion to the Southern Cross (2022 Jan 18)

Post by neufer » Thu Jan 20, 2022 9:45 pm

johnnydeep wrote: Thu Jan 20, 2022 7:44 pm
bystander wrote: Thu Jan 20, 2022 6:49 pm
neufer wrote: Thu Jan 20, 2022 5:48 pm
  • 1489 is a prime number.
    The only proper factors of the square of a prime number p are: 1 & p.

    Hence, [1+p2] => [1 + p]
I don't think that follows. For example, let p = 13

[170] ??? [14}
And more than that, for 1489 in particular, 1+1489*1489=2217122, the factors of that are {2, 1108561}, and the factors of 2+1108561 are {1, 3, 163, 489, 2267, 6801, 369521}, etc. There are some primes of the form 1+p2 but it is a hard - unsolved! - problem to prove that there are infinitely many such primes. See https://en.wikipedia.org/wiki/Landau's_problems

(Factors from https://www.mathsisfun.com/numbers/fact ... .html#calc)
My bad...correction:
  • 1489 is a prime number.
    The only proper factors of the square of a prime number p are: 1 & p.

    Hence, p2=> [1 + p]
-----------------------------------------------------------------------------------------------------------
2362 is a number whose aliquot sequence spirals into the amicable pair: {1184 <=> 1210} like:
  • 2152 => 1898 => {1210 <=> 1184}
    2122 => 1064 => 1336 => {1184 <=> 1210}
    2008 => 1772 => 1336 => {1184 <=> 1210}
    1816 => 1604 => {1210 <=> 1184}
    1690 => 1604 => {1210 <=> 1184}
    1490 => {1210 <=> 1184}
    1420 => 1604 => {1210 <=> 1184}
    1308 => 1772 => 1336 => {1184 <=> 1210}
    1064 => 1336 => {1184 <=> 1210}
[Note: There are certainly more and/or longer "arms"
to the {1184 <=> 1210} "spiral galaxy amicable bar"
...(i.e., 14892 => 1490 => {1210 <=> 1184})
but they must start at numbers larger than 2362.]
------------------------------------------------------------------------------------
To my numerous Starship Asterisk* number theory fans:

What really blows me away about sociable number cycles
is the erratic nature of the following table:
https://en.wikipedia.org/wiki/Sociable_number wrote:
The following categorizes all known sociable numbers as of July 2018
by the length of the corresponding aliquot sequence:

Code: Select all

Cycle length 		Known cycles
--------------------------------------------------
1 (Perfect numbers)	51
2 (Amicable numbers)	1225736919
3 			0
4 			5398
5 			1
6 			5
7 			0
8 			4
9 			1
.
28 			1
How the heck can we understand the universe when this grade school math seems crazy? :roll:
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Re: APOD: From Orion to the Southern Cross (2022 Jan 18)

Post by bystander » Fri Jan 21, 2022 4:04 am

neufer wrote: Thu Jan 20, 2022 9:45 pm
My bad...correction:
  • 1489 is a prime number.
    The only proper factors of the square of a prime number p are: 1 & p.

    Hence, p2=> [1 + p]

    ...
I'm not even sure what that means, again take p=13 ...

169 => [14] ???
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Re: APOD: From Orion to the Southern Cross (2022 Jan 18)

Post by neufer » Fri Jan 21, 2022 1:45 pm

bystander wrote: Fri Jan 21, 2022 4:04 am
neufer wrote: Thu Jan 20, 2022 9:45 pm
My bad...correction:
  • 1489 is a prime number.
    The only proper factors of the square of a prime number p are: 1 & p.

    Hence, p2=> [1 + p]
I'm not even sure what that means, again take p=13 ...

169 => [14] ???
This isn't: 169 = [14].

This IS: 169 => Sum of proper factors 1 & 13 = [14].

It all involves a generalization of the classical concept of:
A perfect number like: 28 => Sum of proper factors 1, 2, 4, 7 & 14 = [28].

In this [sum of proper factors] evolving universe:
  • All primes => [1].
    All perfect numbers => [themselves].
However, many numbers evolve into the "amicable spiral galaxies"
. discussed above...which I think is kinda cool. 8-)
Art Neuendorffer

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bystander
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Re: APOD: From Orion to the Southern Cross (2022 Jan 18)

Post by bystander » Fri Jan 21, 2022 3:31 pm

neufer wrote: Fri Jan 21, 2022 1:45 pm
bystander wrote: Fri Jan 21, 2022 4:04 am
neufer wrote: Thu Jan 20, 2022 9:45 pm
My bad...correction:
  • 1489 is a prime number.
    The only proper factors of the square of a prime number p are: 1 & p.

    Hence, p2=> [1 + p]
I'm not even sure what that means, again take p=13 ...

169 => [14] ???
This isn't: 169 = [14].

This IS: 169 => Sum of proper factors 1 & 13 = [14].

It all involves a generalization of the classical concept of:
A perfect number like: 28 => Sum of proper factors 1, 2, 4, 7 & 14 = [28].

In this [sum of proper factors] evolving universe:
  • All primes => [1].
    All perfect numbers => [themselves].
However, many numbers evolve into the "amicable spiral galaxies"
. discussed above...which I think is kinda cool. 8-)
You are certainly inconsistent with your nomenclature.

You demonstrated [132] = 14, but [14] = 1+2+7 = 10
Know the quiet place within your heart and touch the rainbow of possibility; be
alive to the gentle breeze of communication, and please stop being such a jerk.
— Garrison Keillor

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Re: APOD: From Orion to the Southern Cross (2022 Jan 18)

Post by johnnydeep » Fri Jan 21, 2022 4:33 pm

bystander wrote: Fri Jan 21, 2022 3:31 pm
neufer wrote: Fri Jan 21, 2022 1:45 pm
bystander wrote: Fri Jan 21, 2022 4:04 am

I'm not even sure what that means, again take p=13 ...

169 => [14] ???
This isn't: 169 = [14].

This IS: 169 => Sum of proper factors 1 & 13 = [14].

It all involves a generalization of the classical concept of:
A perfect number like: 28 => Sum of proper factors 1, 2, 4, 7 & 14 = [28].

In this [sum of proper factors] evolving universe:
  • All primes => [1].
    All perfect numbers => [themselves].
However, many numbers evolve into the "amicable spiral galaxies"
. discussed above...which I think is kinda cool. 8-)
You are certainly inconsistent with your nomenclature.

You demonstrated [132] = 14, but [14] = 1+2+7 = 10
"A foolish consistency is the hobgoblin of little minds." Neufer is nothing if not sometimes inconsistent :ssmile:

In this case though, the notation [n] can be taken to be the function that sums the proper factors of n.

So, [13*13] = 1+13 = 14, and [14] = 1+2+7 = 10

At least, that's what we have apparently settled on.
--
"To B̬̻̋̚o̞̮̚̚l̘̲̀᷾d̫͓᷅ͩḷ̯᷁ͮȳ͙᷊͠ Go......Beyond The F͇̤i̙̖e̤̟l̡͓d͈̹s̙͚ We Know."

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Re: APOD: From Orion to the Southern Cross (2022 Jan 18)

Post by neufer » Fri Jan 21, 2022 8:34 pm

johnnydeep wrote: Fri Jan 21, 2022 4:33 pm
bystander wrote: Fri Jan 21, 2022 3:31 pm
neufer wrote: Fri Jan 21, 2022 1:45 pm
This isn't: 169 = [14].

This IS: 169 => Sum of proper factors 1 & 13 = [14].

It all involves a generalization of the classical concept of:
A perfect number like: 28 => Sum of proper factors 1, 2, 4, 7 & 14 = [28].

In this [sum of proper factors] evolving universe:
  • All primes => [1].
    All perfect numbers => [themselves].
However, many numbers evolve into the "amicable spiral galaxies"
. discussed above...which I think is kinda cool. 8-)
You are certainly inconsistent with your nomenclature.

You demonstrated [132] = 14, but [14] = 1+2+7 = 10
"A foolish consistency is the hobgoblin of little minds." Neufer is nothing if not sometimes inconsistent :ssmile:

In this case though, the notation [n] can be taken to be the function that sums the proper factors of n.

So, [13*13] = 1+13 = 14, and [14] = 1+2+7 = 10

At least, that's what we have apparently settled on.
[Square brackets] are just [square brackets] and were never intended to be "the notation."

Rather, it was those arrows: => & <=> that were intended to be "the notation."
https://en.wikipedia.org/wiki/Hobgoblin wrote:
<<Hobgoblins seem to be small, hairy little men who are often found within human dwellings, doing odd jobs around the house while the family is asleep. Such chores are typically small tasks like dusting and ironing. While brownies are more peaceful creatures, hobgoblins are more fond of practical jokes. They also seem to be able to shapeshift, as seen in one of Puck's monologues in A Midsummer Night's Dream. Like other fae folk, hobgoblins are easily annoyed.>>
---------------------------------------------------------------------------------------------------
  • . . . A Midsummer Night's Dream: II, i

    Fairy: Either I mistake your shape and making quite,
    . Or else you are that shrewd and knavish sprite
    . Call'd Robin Goodfellow: are not you he
    . That frights the maidens of the villagery;
    . Skim milk, and sometimes labour in the quern
    . And bootless make the breathless housewife churn;
    . And sometime make the drink to bear no barm;
    . Mislead night-wanderers, laughing at their harm?
    . Those that Hobgoblin call you and sweet Puck,
    . You do their work, and they shall have good luck:
    . Are not you he?


    PUCK: Thou speak'st aright;
    . I am that merry wanderer of the night.
    . I jest to Oberon and make him smile
    . When I a fat and bean-fed horse beguile,
    . Neighing in likeness of a filly foal:
    . And sometime lurk I in a gossip's bowl,
    . In very likeness of a roasted crab,
    . And when she drinks, against her lips I bob
    . And on her wither'd dewlap pour the ale.
    . The wisest aunt, telling the saddest tale,
    . Sometime for three-foot stool mistaketh me;
    . Then slip I from her bum, down topples she,
    . And 'tailor' cries, and falls into a cough;
    . And then the whole quire hold their hips and laugh,
    . And waxen in their mirth and neeze and swear
    . A merrier hour was never wasted there.
    . But, room, fairy! here comes Oberon.
Art Neuendorffer