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Talk:Abundant odd numbers: Difference between revisions
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→On Number Theoretic Tasks: extended to 1e11 , to see, if it works
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:Looks good enough to me. I've updated the Go entry to match. BTW getting a different answer to you for the 1000th abundant odd number. According to my reckoning, 498225 is the 1012th.--[[User:PureFox|PureFox]] ([[User talk:PureFox|talk]]) 22:03, 17 May 2019 (UTC)
:The problem seems to be that you (and also Gerard) are assuming that all the odd numbers will end in 5 when in fact there are a few which end in 1, 3, 7 or 9.
:The first such number is 81081 whose proper divisors are:
:1 + 3 + 7 + 9 + 11 + 13 + 21 + 27 + 33 + 39 + 63 + 77 + 81 + 91 + 99 + 117 + 143 + 189 + 231 + 273 + 297 + 351 + 429 + 567 + 693 + 819 + 891 + 1001 + 1053 + 1287 + 2079 + 2457 + 3003 + 3861 + 6237 + 7371 + 9009 + 11583 + 27027, which sum to 81543. So it's definitely an abundant odd number.
: The other 11 are: 153153, 171171, 189189, 207207, 223839, 243243, 261261, 279279, 297297, 351351 and 459459. --[[User:PureFox|PureFox]] ([[User talk:PureFox|talk]]) 23:47, 17 May 2019 (UTC)
:: Thanks for the heads up concerning odd abundant numbers not ending in '''5'''. -- [[User:Gerard Schildberger|Gerard Schildberger]] ([[User talk:Gerard Schildberger|talk]]) 00:01, 18 May 2019 (UTC)
:: Indeed, you are correct. Poor assumption on my part. --[[User:Thundergnat|Thundergnat]] ([[User talk:Thundergnat|talk]]) 00:29, 18 May 2019 (UTC)
==On Number Theoretic Tasks==
Perhaps we should have a tag "This solution is lame, please make it interesting and remove this tag". RC has too many tasks which are being solved by a loop which factorize a sequence of integers and then print something based on an if condition. Better would be if the author of these tasks indicated an interesting solution in the task description based on number theory. For this task I have added a reference which proves a number of properties of Odd Abundant numbers. 3 might be of interest to this task: there are no Odd abundant numbers with fewer than 3 prime factors; if a number is odd and abundant then so too are all odd multiple of that number; and Odd abundant numbers must satisfy the condition (p1/p1-1)*(p2/p2-1)..(pn/pn-1)>2. So p1=3,p2=5 then p3 must be less than 17 because (3/2)*(5/4)*(13/12)=2.03125 and (3/2)*(5/4)*(17/16)=1.9921875. Errors in early implementations uncovered the smallest Odd abundant number not divisible by 5. Let me consider p1=3, p2=7 and p3=11. (3/2)*(7/6)*(11/10)=1.9250000000000003 so there are no Odd abundant numbers not divisible by 5 with 3 prime factors. So p1=3, p2=7, p3=11, p4=13 -> (3/2)*(7/6)*(11/10)*(13/12)=2.0854166666666667 so this is a good place to start looking. So what is the smallest Odd abundant number not divisible by 3?--[[User:Nigel Galloway|Nigel Galloway]] ([[User talk:Nigel Galloway|talk]]) 21:29, 19 May 2019 (UTC)
:So what is the smallest Odd abundant number not divisible by 3:
<pre>From zumkeller numbers, check every 20/10 th: 5/25/35/55 ...
0: 5391411025 :384 : 5^2*7*11*13*17*19*23*29 _chk_5391411025_SoD_10799308800// sum of divisors
1:26957055125 :512 : 5^3*7*11*13*17*19*23*29 _chk_26957055125_SoD_54344908800
2:28816162375 :512 : 5^3*7*11*13*17*19*23*31 _chk_28816162375_SoD_57967902720
3:33426748355 :512 : 5*7*11*13*17*19*23*29*31 _chk_33426748355_SoD_66886041600
4:34393484125 :512 : 5^3*7*11*13*17*19*23*37 _chk_34393484125_SoD_68836884480
5:37739877175 :576 : 5^2*7^2*11*13*17*19*23*29_chk_37739877175_SoD_76945075200
6:40342627325 :576 : 5^2*7^2*11*13*17*19*23*31_chk_40342627325_SoD_82074746880
7:48150877775 :576 : 5^2*7^2*11*13*17*19*23*37_chk_48150877775_SoD_97463761920
8:50866790975 :576 : 5^2*7^2*11*13*17*19*29*31_chk_50866790975_SoD_102593433600
9:53356378075 :576 : 5^2*7^2*11*13*17*19*23*41_chk_53356378075_SoD_107723105280
10:55959128225 :576 : 5^2*7^2*11*13*17*19*23*43_chk_55959128225_SoD_112852776960
11:59305521275 :576 : 5^2*7*11^2*13*17*19*23*29_chk_59305521275_SoD_119692339200
12:60711976325 :576 : 5^2*7^2*11*13*17*19*29*37_chk_60711976325_SoD_121829702400
13:61164628525 :576 : 5^2*7^2*11*13*17*19*23*47_chk_61164628525_SoD_123112120320
14:63395557225 :576 : 5^2*7*11^2*13*17*19*23*31_chk_63395557225_SoD_127671828480
15:64899009175 :576 : 5^2*7^2*11*13*17*19*31*37_chk_64899009175_SoD_129951682560
16:67275433225 :576 : 5^2*7^2*11*13*17*19*29*41_chk_67275433225_SoD_134653881600
17:68972878975 :576 : 5^2*7^2*11*13*17*19*23*53_chk_68972878975_SoD_138501135360
18:70088343325 :576 : 5^2*7*11*13^2*17*19*23*29_chk_70088343325_SoD_141162393600
19:74922022175 :576 : 5^2*7*11*13^2*17*19*23*31_chk_74922022175_SoD_150573219840
20:75665665075 :576 : 5^2*7*11^2*13*17*19*23*37_chk_75665665075_SoD_151610296320
21:76781129425 :576 : 5^2*7^2*11*13*17*19*23*59_chk_76781129425_SoD_153890150400
22:79383879575 :576 : 5^2*7^2*11*13*17*19*23*61_chk_79383879575_SoD_159019822080
23:87192130025 :576 : 5^2*7^2*11*13*17*19*23*67_chk_87192130025_SoD_174408837120
24:91653987425 :576 : 5^2*7*11*13*17^2*19*23*29_chk_91653987425_SoD_184188211200
25:97974952075 :576 : 5^2*7*11*13*17^2*19*23*31_chk_97974952075_SoD_196467425280
real 76m32,432s
</pre> -[[User:Horsth|Horsth]] 15:20, 19 July 2021 (UTC)
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