[QUOTE=grandpascorpion]
Quartets (using a tweaked version of your code):
[/QUOTE]
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

Corrections
 

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.

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.

And here it is an exe for P4.

[QUOTE=grandpascorpion]Actually, your x=2 answer doesn't fit the original definition.[/QUOTE]
As you can see I used my definition.
[QUOTE]
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.[/QUOTE]
There was no problem for me. But I've rewritten only the gmp version.

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

Thanks for the gmp version.

Smooth Duo List / Duos, Triplets and Quartest submitted to OEIS / A002072 extended
 

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 [url]http://www.research.att.com/~njas/sequences/?q=A002072[/url]
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.

[QUOTE=grandpascorpion;86704]
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.
[/QUOTE]

Thanks! Great work, grandpa.

[QUOTE=grandpascorpion;86704]

On a related note, I found some addition terms for [url]http://www.research.att.com/~njas/sequences/?q=A002072[/url]
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

[/QUOTE]

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.

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

Thanks:smile:

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:
[url]http://www.research.att.com/~njas/sequences/a002072.py.txt[/url]

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

Sequences [B]A002072[/B] and [B]A117581[/B] 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 [I]gpd[/I](n) for greatest prime divisor of n, what is [B][I]S'([/I]23)[/B], the greatest S for which n = S(S-1) has [I]gpd[/I](n) = 23?

The answer of course is 5142501, but this is less than [B][I]S'([/I]23) = [/B]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:

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