Problems & Puzzles: Puzzles

 Puzzle 798. A nice puzzle by Kamenetski. Dmitry Kamenetsky sent the following nice puzze: Let f(n, i) be the sum of n consecutive primes starting from the i-th prime. Let g(n, i) be the number of consecutive terms in the sequence {f(n, i),f(n, i+1),f(n, i+2),...} that are all prime, n>1. For example g(3, 7)=5, because the following 5 sums are all prime: f(3, 7) = p(7)+p(8)+p(9) = 17+19+23 = 59 f(3, 8) = p(8)+p(9)+p(10) = 19+23+29 = 71 f(3, 9) = p(9)+p(10)+p(11) = 23+29+31 = 83 f(3, 10) = p(10)+p(11)+p(12) = 29+31+37 = 97 f(3, 11) = p(11)+p(12)+p(13) = 31+37+41 = 109 Current Kamenetsy's record is g(n, i) = 9 for certain n & i. Q. What is the largest value of g(n, i) that you can find?

Contribution came from Vladimir Chirkov (Set 3, 2015)

All solutions that I found now are:

g(57,754426)=9;
g(41,803278)=9;
g(307,2096506)=9;
g(97,4785746)=9;
g(355,7674053)=9;
g(6979,57208)=9;
g(9179,73999)=9;
g(11791,46206)=9.

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Jeff Heleen wrote (Set 4, 2015)

The earliest instance I found for g(n,i)=9 is

g(41,803278) = 12248837 + … + p(41) =  502214861, prime
12248839 + … + p(41) =  502215517, prime
12248867 + … + p(41) =  502216177, prime
12248879 + … + p(41) =  502216837, prime
12248881 + … + p(41) =  502217491, prime
12248897 + … + p(41) =  502218209, prime
12248917 + … + p(41) =  502218949, prime
12248921 + … + p(41) =  502219673, prime
12248923 + … + p(41) =  502220399, prime

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Vladimir wrote again on Set 8, 2015:

There are the solutions for minimal i then g(n,i) = 1...10 (n<30) in the table

|    |                               g(n,i)=...                              |
|    | 1| 2|  3|   4|    5|     6|       7|        8|          9|          10|
|    +--+--+---+----+-----+------+--------+---------+-----------+------------+
|n= 3| -| -|  3|   -|    7|  5454|   31076|  8744076|  697642916| 23169509240|
|n= 5| -| -|  -|   3| 1547|205058|       -|   992975|  238146201|175025296702|
|n= 7| -| -|  -|   -|    7|     -|   87470|237414878|  520209568| 50284335391|
|n= 9| 2| -| 10|  44|10142|     -|   11141|   931355|11711619785| 71624813079|
|n=11| -| -|  -|   3|   26| 18180|       -|   251887|  752508005|           ?|
|n=13|10|24|  -|   -|    -|   248|       -|  1600214| 5679561065| 62439867642|
|n=15| 2| -|  8|  55|    -|     -|    1387|   813318|  488985942| 27527241232|
|n=17| -| 2| 15| 113|    -|   362|  487432|184316209| 9546021494|           ?|
|n=19| 5| -|  7|1100|20792|756500| 6801560| 45099016|58976442906| 70287268761|
|n=21| 4|16|116|1555|10234|169314|28207232| 40018640|  146171446|159907818552|
|n=23| 4| -| 20| 655| 1649| 21452| 1111783|211257958| 6231562586|166781302853|
|n=25| -| 3|535|   -| 5063|     -|  341240| 78131929|51734332174|           ?|
|n=27| 4| -|  -|   -|   16|  7369|  587682|  1676116|  106368474|100555694594|
|n=29| 6|33| 41|1829|11297|146168| 3967490| 45608324|  866441578| 58115591447|

For example:
g(27,1676116)=8 is the first solutions for n=27 then g(n,i)=8

And also I found a solution for g(n,i)=11:
g(3,29165083170)=11

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On Set 19, 2015 Dimitry Kamenetsky wrote:

I found one more solution that gives a score of 11: g(3,102711011888).

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