Program for searching all odd-16-digit Aliquot cycles

[Stargate38]

Attach your results file. You found an Aliquot cycle.

[legendarymudkip]

Originally Posted by Stargate38

Attach your results file. You found an Aliquot cycle.

I haven't finished the range, and have found a few cycles. This is the only one with the star at the end of the line, so I was wondering if it means something different, and what if it does.

[Drdmitry]

Originally Posted by legendarymudkip

I haven't finished the range, and have found a few cycles. This is the only one with the star at the end of the line, so I was wondering if it means something different, and what if it does.

There are two ways an Aliquot cycle can be found in a program. In the standard way the cycle is found if the number currently under consideration appears to be the minimal element of a cycle. However there can be another situation: the Aliquot sequence started with the given number ends in a cycle. In the second case the line "New cycle found!!*" appears with the star at the end. Sometimes numbers in that cycle can even be bigger than 16 digits.

[Drdmitry]

All my ranges are finished. Here are the results:
For 1: 119 pairs are found, 45 of them new.
For 3 - 7: 376 pairs are found (I added 13 isotopic pairs here), 198 of them new.
For 8 - 9: 138 pairs are found, 59 of them new.

[AndrewWalker]

Drdmitry if you want a copy of the list I sent to Pat send me your email in a private message, it is 4.2 Mb and over 40k pairs.
Do you or anyone else have a program for processing/finding isotopic pairs? Jan used to look for them in submitted pairs so I've never bothered, Pat said his program doesn't check for them. If not I'll write up something myself, I saw your program came a list of the odd substitutions do you have the even substitutions?

Andrew

[Drdmitry]

Originally Posted by AndrewWalker

Drdmitry if you want a copy of the list I sent to Pat send me your email in a private message, it is 4.2 Mb and over 40k pairs.
Do you or anyone else have a program for processing/finding isotopic pairs? Jan used to look for them in submitted pairs so I've never bothered, Pat said his program doesn't check for them. If not I'll write up something myself, I saw your program came a list of the odd substitutions do you have the even substitutions?

Andrew

If you can send me your list in a format similar to that in Pedersen's or Costello's list, that would be great. Then I will be able to find duplicates easily.
I can only detect some of the isotopic pairs (the most popular ones, I hope). If anyone can send me the list of all known small factors of isotopic pairs then I will add them to my detection program. The program itself is not very convenient since I did not design it to be used anywhere else except my computer.
Concerning cycles of even numbers. I did check n up to 100 for 14d cycles and n up to 30 for 15d cycles. However the program I used was not designed for distributed computing. With not too much effort I can make the program for 14d and 15d cycles with the same functionality as the current one. I just do not want to post it here at the moment, because I do not to spread the search too wide. Secondly even numbers are much longer dealt with than odd numbers (maybe 10 times longer). Together with the fact that very few new 14d and 15d even cycles are expected, that search will not be as fruitful as the current one.

[Happy5214]

I'll take 31650000-31659999.

[AndrewWalker]

Just to follow up Pat says he has finished the last from my group. He's going to send me a list of substitutions that he has. I agree searching evens for 15 digits wouldn't result in much, 16 maybe if there are any extra speed ups possible. Have you used any of
the ideas from the papers by te Riele, Moews and others? They use some ideas to show from a partial factorisation of one of the
numbers that sigma() will be outside some range and hence of no use.

te Riele's paper http://www.ams.org/journals/mcom/198...986-0842142-3/ from 1986 is worth a read

Andrew

[Maciej]

I found a dozen or so new pairs of < 10 ^ 15 . I think that even the 100 is still undiscovered .

[legendarymudkip]

Finished 1000-1010, 1010 was done as well, if that isn't the norm because of the program not including the last of a range. 42 cycles found, results attached if I've done it right.

I'll take 30000000-30500000, thanks.

[Drdmitry]

Originally Posted by AndrewWalker

Just to follow up Pat says he has finished the last from my group. He's going to send me a list of substitutions that he has. I agree searching evens for 15 digits wouldn't result in much, 16 maybe if there are any extra speed ups possible. Have you used any of
the ideas from the papers by te Riele, Moews and others? They use some ideas to show from a partial factorisation of one of the
numbers that sigma() will be outside some range and hence of no use.

te Riele's paper http://www.ams.org/journals/mcom/198...986-0842142-3/ from 1986 is worth a read

Andrew

I am looking for aliquot cycles of any length, therefore many of the specific ideas designed for amicable pairs do not work. I use the ideas from Moew's paper about Aliquot cycles.
I split the number into the product d*n where d is the product of small primes and n is coprime with that small primes, Then the most of the products d were ruled out during the preliminary step (mostly because s(d) / d is too small which makes s(d*n) > 2d*n impossible). The most of values d which remain (they are stored in iqs_od_10e16.txt file) already have s(d) / d>2 or very close to 2.
One of the promising ways to speed up the process is to optimize the computation of the sum of divisors (essentially it is the factorization). Currently I use the following procedure: division by small primes, using the value from the precomputed list if the number is small enough, primality check, Pollard rho and finally complete trial division if the rho algorithm fails.