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Old 2015-08-13, 11:39   #23
AndrewWalker
 
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Thanks for the update Sergei! If it's easy to work out it would be interesting to know
what the highest prime factor for the first number is for each digit range.

It would also be interesting to know if any more not divisible by 2 or 3 turn up, after the 16d pair
the next two are in the 19d file.

Andrew

Last fiddled with by AndrewWalker on 2015-08-13 at 11:39
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Old 2015-08-14, 07:15   #24
Sergei Chernykh
 
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Quote:
Originally Posted by AndrewWalker View Post
Thanks for the update Sergei! If it's easy to work out it would be interesting to know
what the highest prime factor for the first number is for each digit range.

It would also be interesting to know if any more not divisible by 2 or 3 turn up, after the 16d pair
the next two are in the 19d file.

Andrew
I haven't found any new pairs coprime to 6 so far. As for largest prime factors:

c2_3.txt: 1 pairs, largest factor 11
c2_4.txt: 4 pairs, largest factor 251
c2_5.txt: 8 pairs, largest factor 137
c2_6.txt: 29 pairs, largest factor 2393
c2_7.txt: 66 pairs, largest factor 60659
c2_8.txt: 128 pairs, largest factor 539783
c2_9.txt: 350 pairs, largest factor 3041279
c2_10.txt: 841 pairs, largest factor 5118431
c2_11.txt: 1913 pairs, largest factor 141374879
c2_12.txt: 4302 pairs, largest factor 974705471
c2_13.txt: 9877 pairs, largest factor 18510661889
c2_14.txt: 21855 pairs, largest factor 152504187263
c2_15.txt: 47728 pairs, largest factor 895732991999
c2_16.txt: 87558 pairs, largest factor 7160665580639
c2_17.txt: 4873 pairs, largest factor 631052035696799
c2_18.txt: 4169 pairs, largest factor 319606267871249
c2_19.txt: 4290 pairs, largest factor 5254683154329599
c2_20.txt: 4742 pairs, largest factor 31159681634452799
c2_21.txt: 4859 pairs, largest factor 477095265966920831
c2_22.txt: 4701 pairs, largest factor 722471849465034239
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Old 2015-08-20, 13:10   #25
Drdmitry
 
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At last the search of all 15d odd cycles is over. In total there are 6 such pairs.
The search of 16d odd cycles had been started on this forum a couple of months ago and is moving slowly. It did not find any 16d odd cycle yet.
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Old 2015-09-11, 12:25   #26
Sergei Chernykh
 
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The search of amicable pairs up to 10^17 is almost finished (currently searching largest prime factors near 10^11). Now that I have more data than ever, I've tried to find a good approximation for A(x) - the number of amicable pairs with smaller member <= x. I know A(x) for x <= 10^16 and a very accurate estimate for x=10^17 based on pairs I found so far and pairs known before.

It can be clearly seen from the table below that A(x)=x1/(3+e(x)) where e(x)->0 as x->infinity.

So I tried f(x)=x1/(3+1/ln(x)) and it seems to be a very good approximation

Code:
x	ln(x)/ln(A(x))	A(x)	f(x)
10^4	5,722706232	5	19
10^5	4,488558588	13	41
10^6	3,696289926	42	89
10^7	3,442469864	108	193
10^8	3,371385028	236	416
10^9	3,251565357	586	896
10^10	3,170150901	1427	1930
10^11	3,121677483	3340	4159
10^12	3,090229261	7642	8960
10^13	3,063502173	17519	19302
10^14	3,046651059	39374	41582
10^15	3,036419958	87102	89579
10^16	3,030003782	190775	192979
10^17	3,025652216	~415550	415737
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Old 2015-09-29, 20:38   #27
Sergei Chernykh
 
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The search of all amicable pairs up to 10^17 completed today at 12:47:01 CEST. There are 415523 pairs with smaller member < 10^17, 224748 of them are 17-digit. Started the search up to 10^18 and found over 20000 new 18-digit pairs in first 10 hours One of those 18-digit pairs is coprime to 6:
Code:
936789193264049125=5^3*7^2*11*13*17*19*23*47*67*131*349
1009532413277262875=5^3*7^2*11*13*17*19*79*659*68543
This pair was submitted to Patrick. New 17-digit pairs haven't been submitted yet, because I'm waiting for him to sort out all 16-digit pairs I submitted more than 2.5 months ago
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Old 2015-10-01, 05:23   #28
R. Gerbicz
 
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Quote:
Originally Posted by Sergei Chernykh View Post
The search of all amicable pairs up to 10^17 completed today at 12:47:01 CEST. There are 415523 pairs with smaller member < 10^17, 224748 of them are 17-digit. Started the search up to 10^18 and found over 20000 new 18-digit pairs in first 10 hours One of those 18-digit pairs is coprime to 6:
Code:
936789193264049125=5^3*7^2*11*13*17*19*23*47*67*131*349
1009532413277262875=5^3*7^2*11*13*17*19*79*659*68543
This pair was submitted to Patrick. New 17-digit pairs haven't been submitted yet, because I'm waiting for him to sort out all 16-digit pairs I submitted more than 2.5 months ago
Congrats!

Returning to speed:
With my new code I can find amicable pairs up to 1e13 in less than 8 hours on a single(!) Core-i3-2350M (2.30 GHz), so faster than your program's time previously posted (25 core hours on Core i7-4770K). It is a c++ code, what it still needs to make a parallel version of this (but that would not be hard with OpenMp, that is available on gcc). We could do a distributed search, but as my program is rather long and the complexity isn't that obvious first test it for smaller limits (say for limit=10^n for n<=16) to see the speed of the code.

ps. I would advice to submit the pairs to Patrick.
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Old 2015-10-01, 05:36   #29
Sergei Chernykh
 
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Quote:
Originally Posted by R. Gerbicz View Post
Congrats!

Returning to speed:
With my new code I can find amicable pairs up to 1e13 in less than 8 hours on a single(!) Core-i3-2350M (2.30 GHz), so faster than your program's time previously posted (25 core hours on Core i7-4770K). It is a c++ code, what it still needs to make a parallel version of this (but that would not be hard with OpenMp, that is available on gcc). We could do a distributed search, but as my program is rather long and the complexity isn't that obvious first test it for smaller limits (say for limit=10^n for n<=16) to see the speed of the code.

ps. I would advice to submit the pairs to Patrick.
25 core hours was 3 months ago - my latest code does 10^13 in 1h 25m on Core i7-4770K (~6.3 core hours). Otherwise I wouldn't be able to finish 10^17 that quick. Still, your program seems to be super fast considering that it runs in the same time on a slower CPU. I'd love to submit all my new pairs to Patrick (it's almost 400000 pairs now), but his software is not good enough yet. He can only process files containing < 500 pairs each, so submitting 800 files via the web form doesn't inspire me...

Last fiddled with by Sergei Chernykh on 2015-10-01 at 06:35
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Old 2015-10-01, 08:35   #30
AndrewWalker
 
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Quote:
Originally Posted by Sergei Chernykh View Post
The search of all amicable pairs up to 10^17 completed today at 12:47:01 CEST. There are 415523 pairs with smaller member < 10^17, 224748 of them are 17-digit. Started the search up to 10^18 and found over 20000 new 18-digit pairs in first 10 hours One of those 18-digit pairs is coprime to 6:
Code:
936789193264049125=5^3*7^2*11*13*17*19*23*47*67*131*349
1009532413277262875=5^3*7^2*11*13*17*19*79*659*68543
This pair was submitted to Patrick. New 17-digit pairs haven't been submitted yet, because I'm waiting for him to sort out all 16-digit pairs I submitted more than 2.5 months ago
Well done with finding them, however I hope everyone can hold back on submissions until he's processed their existing submissions
to make life easier. Processing them with the existing software looks to be time consuming, he said it was only designed for small updates.

I'm also in the position of having older pairs still to submit but I'm not going to flood him with them.

Jan used to be able to take larger submissions, perhaps someone on the board with decent programing experience
could offer to help out? (Pat did mention the program was written by someone who was close to retiring so
obviously having enough spare time is one factor)

Andrew
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Old 2015-10-01, 08:44   #31
Sergei Chernykh
 
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I can at least have a look at his programs and estimate the work needed. Even rewriting it from scratch would take a week max. It's just a database, submission form, download form and a verification program.

Last fiddled with by Sergei Chernykh on 2015-10-01 at 08:49
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Old 2015-10-02, 21:28   #32
Drdmitry
 
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Quote:
Originally Posted by AndrewWalker View Post
Well done with finding them, however I hope everyone can hold back on submissions until he's processed their existing submissions
to make life easier. Processing them with the existing software looks to be time consuming, he said it was only designed for small updates.

I'm also in the position of having older pairs still to submit but I'm not going to flood him with them.

Jan used to be able to take larger submissions, perhaps someone on the board with decent programing experience
could offer to help out? (Pat did mention the program was written by someone who was close to retiring so
obviously having enough spare time is one factor)

Andrew
The submission process indeed looks time consuming. I agree, it should not be a problem to make it faster. I will try and ask Pat if I can help.

By the way, Sergei, maybe you should email Pat and ask him about your submission? He processed my one month old submission which seems to be made later than yours.
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Old 2015-10-02, 21:37   #33
Sergei Chernykh
 
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Quote:
Originally Posted by Drdmitry View Post
The submission process indeed looks time consuming. I agree, it should not be a problem to make it faster. I will try and ask Pat if I can help.

By the way, Sergei, maybe you should email Pat and ask him about your submission? He processed my one month old submission which seems to be made later than yours.
He seems to ignore all my submissions, at least he e-mailed to me that "I have put off breaking up your remaining files" after adding some of my 16-digit pairs in August. Anyway, I'm tired of waiting. I'm creating my own web page for amicable pairs database. It'll be ready this weekend. Submission form already works and I've uploaded all pairs up to 60 digits from Pat's site. There will be no restrictions on file size and submissions will be fully automatic
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