Problems & Puzzles: Puzzles

Puzzle 463. k.P(k)+(k+1).P(k+1) JM Bergot sent the following nice puzzle: It is noticed that 6*P(6) + 7*P(7)=6*13 + 7*17 = 197 prime 7*P(7) + 8*P(8)= 7*17 + 8*19 = 271 prime 8*P(8) + 9*P(9)=8*19 + 9*23 = 359 prime 9*P(9) + 10*P(10)= 9*23 + 10*29 =497 FAIL  There were three successful steps using four consecutive primes from P(6) to P(9). Can you find a starting prime to give five steps...or more? Note: The example given by Bergot can be compressed using two numbers: 3 & P(6); the successful steps & the smaller prime used. Well, I have gotten a solution: 7, P(321150). So the real question is to get a larger than 7 successful steps

---

Contributions came from Enoch Haga and Farideh Firoozbakht.

***

Enoch wrote:

Had my program running for two days without getting the 8 solution. I did verify yours.

To fill in, first occurrence of sequence of:

2 -- 5,7 - 7,11
3 -- 13,17 - 19,23
4 -- 61,67 - 73,79
5 -- 2153,2161 -- 2207,2213
6 -- 2153,2161 -- 2213-2221
7 -- 4580041,4580077 -- 4580201,4580209

***

Farideh wrote:

I haven't found a number with 8 or more successful steps.

All numers with 7 successful steps up to 372*10^6 are:
321150, 31757516, 134558368, 159849354 & 178323284.

***