Triple Double
For those of you who know basketball, you will be familiar with the
concept of the "triple double", that is more than 10 of 3 separate categories of feats within a single game of basketball. (points, rebounds, assists, if I remember correctly) Some k values of the series k*2^n+1 and k*2n1 will also achieve this, but maybe they are rarer. The feats are: 10 twins, 10 Sophie Germain primes and 10 Cunningham chains length 2 first kind, with longer chains, say Cunningham chains of length 3, counting as two separate chains. In fact "Triple doubles" are proving slightly elusive. After about 4 days of trying the closest I have achieved is k= 20943627705 Which has 9 SGs, 8 CC first kind and 10 twins, checked to n=2000 9+8+10=27 close to my best points score to date of 28 Another close one: k= 92052927781815 With 7, 9 and 9 respectively, also checked to n=2000 My search has been on payam numbers E(27) and I checked quite a lot. Can you do better in this all round category? Regards Robert Smith 
Progress
I am a bit frustrated, as the triple double appears to be a very hard challenge. So far I have spent about three computer weeks on trying to find a triple double, and, to date, cannot beat 28 points, let alone the minimum 30 points required for the triple double.
I am using Axn1's program which generates "double payam" numbers, that is, those numbers which do not have small factors for either power series k.2^n+1 and k.2^n1. I have been running the sieve to generate such k which are guaranteed to have no factors smaller than 19 for any value of n, and then further sieving out all k which have a factor smaller than 256 for n from 1 to 10, + or , again using Axn's "Payamx" program. In this way, I have discovered Cunningham Chains of length 10, both + and , but so far, none which are chains of 10 for both + and . If this was the case, we would have nine chain points for each of the + and  and 10 twins. Given that the longest bitwin known is not 10 segments in length, it is optimistic to expect that I would have discovered a k with such properties. However, I might have expected, say, a Cunningham Chain of the first kind length 10 (worth 9 points) and, say, 5 CCs of the second kind, and hence 6 twins, a score of 20 points seemed very feasible. In fact, I have checked for all such values of k up to approx 1.6*10^20, and the number of k which satisfy the sieve criteria are approximately 40 million. I have run all of these 40 million values up to n=10, and I have found no value with greater than 17 "triple double" points, a rather disappointing maximum. I have run the approximately 200,000 values which have 7 points or greater up to n=100, and those which had more than 13 points at 100 are taken further up to 500. Finally, any with more than 20 points at this stage up to k=4000. I obtained about 20 values with 24 or more points including one new one with 28, and 2 with 27. In addition I have found only 5 candidates which qualify as "double doubles", that is usually 10 or more twin primes and 10 or more either CC type 1 or 2. It is worthwhile noting that, using these k, that the point score for CC's tend to be higher than the pooint scores for twins. I have found plenty of k with 12, 13, 14 and even 15 CCs, but no k with more than 12 twins. Regards Robert Smith 
Solved!!!!!
For the last two years I set myself a formidable target, to find a k such that there are 10 or more twins, Cunningham chains of the second kind and Sophie Germains. The challenge is called the triple double, after the equivalent (but easier) basketball feat.
Finally, after a concerted effort during the last week, I can announce the first such k discovered, 13007751179860962525, and I consider myself lucky to find it. The value have the following 10 Sophie Germain pairs: [3,4], [4,5], [12,13], [28,29], [90,91], [187,188], [415,416], [487,488], [587,588], [889,890] 14 Cunningham chains of the second kind: [2,3], [3,4], [4,5], [5,6], [6,7], [26,27], [32,33], [41,42], [68,69], [100,101], [131,132], [154,155], [454,455] And 12 twins: 3, 4, 5, 7, 12, 17, 26, 29, 33, 42, 69, 155 I have taken n to 10000 without any more pairs of interest. The k value is a deficient primorial of the form 23#/34 and is a t12 double payam number. I found the value by searching for twins, which are most difficult form of the three, amongst t12 double payams, I checked many millions of t12 values, with a preliminary sieve to only further process those with no factors of less than 257 in the first 6n, and finding those with at least 3 twins at n=6, 7 twins at n=100, and 9 twins at n=500. Of the 185 values with 10 or more twins, only two values gave 10 or more SGs and 4 gave 10 or more CC2s. For this reason I consider it lucky that one value was in both camps. I would like to give thanks at “Axn1” for his fast double payam number generator, and to “Citrix” for his fast twin counter. Values were rechecked for prp3 using pfgw Regards Robert Smith 
It was long in coming but worth it. WTG!

Congrats! What an example of perseverance ! :bow:
(For novices it would be nice to know an explicit formula linking the small numbers to the big number....) Also, nice Cunningham chain of length 6  and you would have qualified even by counting it only as 1! Retrospectively, wouldn't that have been an efficient sieving criterion ? (By requiring to have a chain of length 6, you can advance in steps of 6 when scanning nvalues. But I don't know if one could expect a candidate with the chain [2,3,4,5,6,7]  what's the relative frequency of such among "good candidates" ?) 
Wow! Congratulations on the find.
Now the next question is: Is there a k < 13007751179860962525 with the triple double property? What is the smallest such k? :popcorn: 
[QUOTE=m_f_h;107422]Congrats! What an example of perseverance ! :bow:
(For novices it would be nice to know an explicit formula linking the small numbers to the big number....) Also, nice Cunningham chain of length 6  and you would have qualified even by counting it only as 1! Retrospectively, wouldn't that have been an efficient sieving criterion ? (By requiring to have a chain of length 6, you can advance in steps of 6 when scanning nvalues. But I don't know if one could expect a candidate with the chain [2,3,4,5,6,7]  what's the relative frequency of such among "good candidates" ?)[/QUOTE] For some reason, never satisfactorily explained, twins are less common than CCs, evidenced by the records of 15 twins and 18 CCs. So tacking twins is the most important thing to do to find a triple double. It is fiendishly difficult to find long bitwin chains, but you are right it is not a bad way to try to find a good base on which to build hoping that it would also provide twins etc. at higher n. In fact I tried this approach, but using mods to build such k tends to overlook the fact that at the next few levels of n, then the values will almost all be composite, which is why I went for a more balanced approach of finding k's with a good number twins in the first few n, and then testing those further After the t12 discovery, I checked a great number of t9s, but no triple double. Regarding frequencies: 1. t9 double payams occur with a frequency of 1 in every 1,875,000 odd k 2. t9 double payams that have no factors smaller than 257 in the first 7 n, plus or minus are 1 in 10,500 t9 double payams 3. In a test run of the first 53,256,000 t9s with the property in 2, the probabilities of getting 4,5,6,7 twins in the first 7 n are: 4 twins  1 in every 1,667 tested 5 twins  1 in every 37,086 tested 6 twins  1 in every 873,052 tested all 7 twin  only one found in that range. Obviously for higher k, these values drop off quickly. Hope this helps 
[QUOTE=masser;107424]Wow! Congratulations on the find.
Now the next question is: Is there a k < 13007751179860962525 with the triple double property? What is the smallest such k? :popcorn:[/QUOTE] I would bet that there is a smaller value, and maybe a number of them, but there are a lot of k to check!!!! 
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