Problems & Puzzles: Puzzles

Puzzle 862. Redo Puzzle 861 Redo Puzzle 861, including the arithmetic multiplication operation to the two previous accepted operations (adding and subtracting). Example: For k=3, N={2, 3, 7}, n=9 2=2 3=3 5=2+3 7=7 11=2*7-3 13= 2*3+7 17=2*7+3 19=3*7-2 23=3*7+2 Q=n/k=9/3=3. I'm not sure if this example is optimum for k=3.  Q. Send your solution with your best Q value for k=2, 3, 4, ... 10.

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Contributions came from Dmitry Kamenetsky, Michael Hürter and Emmanuel Vantieghem

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Dmitry wrote:

{2, 6, 11} gives 9 consecutive primes with Q=3
 

{3, 4, 10, 19} gives 26 consecutive primes with Q=6.5
 

{6, 12, 13, 14, 17} gives 58 consecutive primes with Q=11.6
 

{5, 8, 19, 21, 25, 31} gives 140 consecutive primes with Q=23.3
 

{12, 24, 27, 30, 34, 35, 37} gives 339 consecutive primes with Q=48.4

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Michael wrote:

I have the following results:

For k = 3 I found n = 9 too.
k = 4: n = 26    10 19 4 3
k = 5: n = 58    18 14 13 6 11

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Emmanuel wrote:

I listed : k, N, the number of primes produced, the biggest prime produced and Q : 

 

3, {2, 3, 7}, 9, 23, 3

4, {3, 4, 10, 19}, 26, 101, 13/2

5, {5, 6, 11, 13, 18}, 58, 271, 58/5

6, {6, 17, 25, 29, 30, 36},188,, 1123, 94/3

7, {2, 6, 18, 54, 188, 530, 965}, 305, 2011, 305/7

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