Problems & Puzzles: Puzzles

Puzzle 159.  A follow up to Puzzle 37

Daniel Gronau wrote some weeks ago:

Here is an extension of puzzle 37:

Find a large set of K distinct numbers {a(k)}, where 2a(i)a(j) + 1 is prime for i<>j, in the least range(*) possible. Do this for K = 10, 11, 12, ...

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(*) the range is a(k)-a(1)


Solution:

Anurag Sahay found (Set. 2, 05) the following solution for K=10:

k=10: { 87,450,189,192,357,2242,27075,28269,103707,671205 }

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C. Rivera found (Set. 4, 05) the following two 'small-range' solutions:

K=9:

{3 6 26 45 168 195 243 348 385}, perhaps this is the minimal range...
 

K=10:

{10 51 60 81 165 294 1050 1199 1659 3129}, can you get a smaller range?

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The 9/9/05 Giovanni Resta reported the following results:

K GAP VALUES
----------------------------
2 [1] 1, 2
3 [2] 1, 2, 3
4 [5] 1, 3, 5, 6
5 [18] 3, 11, 16, 18, 21
6 [42] 6, 13, 33, 35, 36, 48
7 [95] 1, 3, 6, 33, 56, 63, 96
8 [164] 1, 3, 11, 30, 51, 81, 105, 165
9 [382] 3, 6, 26, 45, 168, 195, 243, 348, 385
10 [2580] 27, 30, 153, 195, 360, 612, 945, 1353, 2243, 2607
11 [12945] 9, 30, 147, 429, 592, 737, 4650, 4845, 5202, 7182, 12954
12 [267890] 1, 9, 30, 2109, 3474, 4026, 5144, 11844, 22059, 23430, 242229, 267891

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Wilfred Whiteside wrote:

Sadly, my computer must be shut down.  Hurricane Rita, the witch from hell is headed our way.  It's wind speed has passed through one too many prime numbers for my liking, so we are going to evacuate tomorrow.  Found a k=12 to report but plan to crank a bit more post Rita if my brand new dual core machine survives.  So far found:
 
K=12, [score=57408]
 
27,77,120,855,1407,2604,5782,6894,8085,54327,54384,57435
 
produces 66 unique primes for 2*xi*xj+1, i<>j
 
Windows - Shutting Down and Boarding Up

So, the Giovanni's record for k=12 has been improved!...

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