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
[quote=johnnydeep post_id=319981 time=1642613872 user_id=132061]
[quote=neufer post_id=319978 time=1642607541 user_id=124483]
[quote=Ann post_id=319949 time=1642486563 user_id=129702]
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.[/quote][quote=https://en.wikipedia.org/wiki/Amicable_numbers]
[list][b]factors of 2362 = 1, 2, and 1181
[color=#0000FF]. [1+2+1181 = 1184]
factors of 1184 = 1, 2, 4, 8, 16, 32, 37, 74, 148, 296, and 592[/color]
[color=#FF0000]. [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[/color]
[color=#0000FF]. [1+2+5+10+11+22+55+110+121+242+605 = 1184][/color][/b][/list][/quote][/quote]
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.[/quote]
It's a number whose aliquot sequence spirals into the amiable pair: [color=#FF0000]{1184 <=> 1210}[/color] like:
[list][b][color=#0000FF] 2152 => 1898[/color] => [color=#FF0000]{1210 <=> 1184}[/color]
[color=#0000FF] 2122 => 1064 => 1336[/color] => [color=#FF0000]{1184 <=> 1210}[/color]
[color=#0000FF] 2008 => 1772 => 1336[/color] => [color=#FF0000]{1184 <=> 1210}[/color]
[color=#0000FF] 1816 => 1604[/color] => [color=#FF0000]{1210 <=> 1184}[/color]
[color=#0000FF] 1690 => 1604[/color] => [color=#FF0000]{1210 <=> 1184}[/color]
[color=#0000FF] 1490[/color] => [color=#FF0000]{1210 <=> 1184}[/color]
[color=#0000FF] 1420 => 1604[/color] => [color=#FF0000]{1210 <=> 1184}[/color]
[color=#0000FF] 1308 => 1772 => 1336[/color] => [color=#FF0000]{1184 <=> 1210}[/color]
[color=#0000FF] 1064 => 1336[/color] => [color=#FF0000]{1184 <=> 1210}[/color][/b][/list]
[Note: There are certainly more and/or longer [b][color=#FF00FF]"[i]arms[/i]"[/color][/b] to the [b][color=#FF0000]{1184 <=> 1210}[/color] [color=#FF00FF]"[i]spiral galaxy[/i]"[/color][/b]
(i.e., [b][color=#0000FF][1+1489[sup]2[/sup]] => 1490[/color] => [color=#FF0000]{1210 <=> 1184}[/color][/b])
but they must start at numbers larger than 2362.]
[quote=https://oeis.org/A121508]
Conjectured list of numbers whose aliquot sequence eventually reaches a
[b][color=#FF0000]sociable number cycle[/color] of length two or more, but [u][color=#0000FF]which are not themselves part of the cycle[/color][/u][/b].
[b][color=#0000FF]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...[/color][/b][/quote][quote=https://en.wikipedia.org/wiki/Sociable_number]
<<In mathematics, [b][color=#FF0000]sociable numbers[/color][/b] 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&aq=2678&action=last20&fr=0&to=40[/quote]