googology...?
I can find a few examples of number sequences that start late.
- Carmichael numbers start at 561.
- Taxicab numbers start at 1729.
- Pólya conjecture fails at 906150257.
FB100Z • talk • contribs 23:42, July 22, 2013 (UTC)
I think Pólya conjecture deserves an article. It shows that no small counterexamples doesn't imply truth of hypothesis. I also like following example: I state that all numbers are smaller than million. After checking hundreds of thousands of numbers I found no counterexamples, so statement seems to hold. LittlePeng9 (talk) 07:19, July 23, 2013 (UTC)
More pages about abundant numbers?[]
- 27720 is the first abundant number whose factors have a sum greater than 4 times the original number.
- 122522400 is the first abundant number whose factors have a sum greater than 5 times the original number.
- 130429015516800 is the first abundant number whose factors have a sum greater than 6 times the original number.
- Source: OEIS
- 20821017304425168561312837502762890375 is the first odd-abundant number whose factors have a sum greater than 4 times the original number.
- Source: own research
- Should we create these?
- Should we create category: Abundant numbers?
Wythagoras (talk) 16:08, July 26, 2013 (UTC)
I can find more numbers, but should we create these? Wythagoras (talk) 14:41, August 3, 2013 (UTC)
If we created pages for these numbers, I think we should limit ourselves to odd-abundant numbers, as they give more extreme bounds. LittlePeng9 (talk) 20:56, August 3, 2013 (UTC)
Okay, here is more: 48870871124826570463953805139878697155358000962012333290725030523875 is the first odd-abundant number not divisible by 3 whose factors have a sum greater than 3 times the original number. And how about perfect numbers? We have already 6, 28 and 496, so why not 8128 ( = the 4th perfect number ) and 33550336 ( = the 5th perfect number ) Wythagoras (talk) 06:36, August 4, 2013 (UTC)
10^n+1 never abundant?[]
What is the smallest n, such that 10n + 1 is an abundant number? --84.61.140.84 08:41, April 17, 2017 (UTC)
Growth rate[]
Let f(x,y) be smallest x-rough number whose factors has a sum of more than y times the original number. So
- f(3,2) = 945
- f(3,3) = 1018976683725
- f(3,4) = 20821017304425168561312837502762890375
- f(5,2) = 5391411025
- f(5,3) = 48870871124826570463953805139878697155358000962012333290725030523875
- f(13,2) = 7970466327524571538225709545434506255970026969710012787303278390616918473506860039424701
In general, what is the growth rate of f in terms of x and y, respectively? {hyp/^,cos} (talk) 10:14, 26 January 2022 (UTC)
Added a reference[]
I have added reference OIES in description. --Gongxiang01 (talk) 09:15, 1 March 2024 (UTC)