Smooth power trios

[R. Gerbicz]

Originally Posted by grandpascorpion

Quartets (using a tweaked version of your code):

I also modified my program. These are the smallest smoothquartets solutions.

x=2

1680=2^4*3*5*7
1681=41^2
1682=2*29^2
1683=3^2*11*17

strength=log(1681)/log(41)=2
==========================================
x=3

3678723=3^3*19*71*101
3678724=2^2*7^2*137^2
3678725=5^2*37*41*97
3678726=2*3*83^2*89

strength=log(3678723)/log(101)=3.27577
==========================================
x=4

22377473780=2^2*5*139*179*193*233
22377473781=3*13*43*103*353*367
22377473782=2*19^3*67*97*251
22377473783=7*29*31*107*167*199

strength=log(22377473781)/log(367)=4.03554

[grandpascorpion]

Actually, your x=2 answer doesn't fit the original definition.

I'm wrong about x=3 and x=4. I thought those answers weren't smooth enough. Which leads me to, when you were running for x=3, did you get a false positive with 3027675. I did when I ran it for 25 3.

[R. Gerbicz]

One really large 74 bits smoothtrio for x=5 found by the smoothtriosgmp program:

16692872914488219204860=2^2*5*13*3853*6133*6299*20639*20899
16692872914488219204861=3^28*7^2*1579*9431
16692872914488219204862=2*97*167*277*563*12841*14951*17209

Now it is saving the found smoothtrios and you can continue the program because it is saving the program's d,x,n value after every 1000000-th iteration. If you want to continue the computation then don't give d and x, because if the number of the input pararmeters isn't 2 then the program will use the smoothtriosstat.txt file.

But note that this gmp version is about 15% slower than smoothtrio.c for d<=62.

You can see the source in the attachment.

[R. Gerbicz]

And here it is an exe for P4.

[R. Gerbicz]

Originally Posted by grandpascorpion

Actually, your x=2 answer doesn't fit the original definition.

As you can see I used my definition.

I'm wrong about x=3 and x=4. I thought those answers weren't smooth enough. Which leads me to, when you were running for x=3, did you get a false positive with 3027675. I did when I ran it for 25 3.

There was no problem for me. But I've rewritten only the gmp version.

[grandpascorpion]

That's fine but it's not really a valid answer.

Thanks for the gmp version.

[grandpascorpion]

Hi R.,

I tweaked your code to check duos and found min answers through the 9th power/term. I decided to use your variant of the definition and submitted all three lists to OEIS, citing your program.

Solutions and factorizations:
3=3,2=2

9=3^2, 8= 2^3

2401=7^4, 2400=2^5*3*5^2 (3rd and 4th term)

5909761=11^2*13^2*17^2, 5909760=2^8*3^5*5*19

1611308700=2^2*3^6*5^2*23*31^2, 1611308699=7^4*11*13^2*19^2

421138799640=2^3*3^5*5*13^4*37*41, 421138799639=17*19*23^2*31*43^3

2286831727304145=3^15*5*7^3*19*67*73, 2286831727304144=2^4*17*23^2*37*41^2*59*61*71

3948741978036988496=2^4*7^5*13*23*43*59^3*67*83, 3948741978036988495=5*11*17*31*97^2*101*103*109*113^2

I submitted these. I decided to use your variant of the definition.


=================================================

On a related note, I found some addition terms for http://www.research.att.com/~njas/sequences/?q=A002072
using a modified version of your program. This list (n and n+1) takes extends this list up to prime = 97. There's no counterexamples < 2^62.

31887350832896
31887350832896
119089041053696
2286831727304144
2286831727304144
17451620110781856
166055401586083680
166055401586083680


Incidentally, for duos, the max value log(n)/log(max prime) was 9.287 for the pair below:

4108258965739505499=3^7*13*19*23^4*47*73*89^2
4108258965739505500=2^2*5^3*7^2*11*29^2*31^2*43^2*101^2

10 would be quite a challenge I think.

[R. Gerbicz]

Originally Posted by grandpascorpion

I tweaked your code to check duos and found min answers through the 9th power/term. I decided to use your variant of the definition and submitted all three lists to OEIS, citing your program.

Thanks! Great work, grandpa.

[Citrix]

Originally Posted by grandpascorpion

On a related note, I found some addition terms for http://www.research.att.com/~njas/sequences/?q=A002072
using a modified version of your program. This list (n and n+1) takes extends this list up to prime = 97. There's no counterexamples < 2^62.

31887350832896
31887350832896
119089041053696
2286831727304144
2286831727304144
17451620110781856
166055401586083680
166055401586083680

Grandpa,

Could I have a copy of the modified program. Is it possible to look for smooth consecutive pairs, using the solution of pell equations, than sieving all numbers and then finding smooth numbers.

http://en.wikipedia.org/wiki/Stormer%27s_theorem

Thanks

[grandpascorpion]

Hi Citrix,

Thanks for this link. To be frank, my script (originally R. Gerbicz's) has a totally different algorithm.

Don Reble has already written a script (which I assume is much more efficientwith Pell approach in Python:
http://www.research.att.com/~njas/se...a002072.py.txt

It would be a good little exercise to convert this to a C program.

[Jim White]

Sequences A002072 and A117581 are the same, only the latter uses the higher value of each pair as the sequence value.

This is preferable, I think, since the only published tables, the 1964 results of Dick Lehmer, use that convention, and his proofs are given in the same terms.

Both sequences have the same inconvenient feature, though - those pairs of duplicate values might preserve the monotonicity (if that's a word!) of the sequence, but only by omitting useful information.

Writing gpd(n) for greatest prime divisor of n, what is S'(23), the greatest S for which n = S(S-1) has gpd(n) = 23?

The answer of course is 5142501, but this is less than S'(23) = 11859211, so it gets left behind!

Anyway, here is a list that fills in those entries, and which extends the sequence to the 35th prime:

N pN S'(pN) log2(S')
=============================================
 1.   2                        2   1
 2.   3                        9   3.1699
 3.   5                       81
 4.   7                     4375
 5.  11                     9801
 6.  13                   123201
 7.  17                   336141
 8.  19                 11859211  23.4995
 9.  23                  5142501  22.2940
10.  29                177182721  27.4077
11.  31               1611308700  30.5856 
12.  37               3463200000  31.6895
13.  41              63927525376  35.8957
14.  43             421138799640  38.6155
15.  47            1109496723126  40.0103
16.  53            1453579866025  40.4027
17.  59           20628591204481  44.2297
18.  61           31887350832897  44.8580
19.  67           12820120234376  43.5435
20.  71          119089041053697  46.7090
21.  73         2286831727304145  51.0223
22.  79      9591468737351909376  63.0565 
23.  83        17451620110781857  53.9542 
24.  89       166055401586083681  57.2044
25.  97        49956990469100001  55.4715
26. 101      4108258965739505500  61.8332
27. 103  19316158377073923834001  74.0322 
28. 107       386539843111191225  58.4234
29. 109     90550606380841216611  66.2954 
30. 113    205142063213188103640  67.4752
31. 127     53234795127882729825  65.5290
32. 131   4114304445616636016032  71.8011
33. 137 124225935845233319439174  76.7173
34. 139   3482435534325338530940  71.5606
35. 149   6418869735252139369210  72.4428