Problems & Puzzles: Puzzles

Puzzle 799. Follow-up to Puzzle 798. Dmitry Kamenetsky sent the following follow-up to his own puzzle 798: Let f(n,i) be the sum of n consecutive primes starting from the i-th prime. Let h(n, i) be the number of consecutive terms in the sequence {f(n, i),f(n+2, i),f(n+4, i),...} that are all prime. For example h(1, 10)=5, because the following 5 sums are all prime:   f(1, 10) = p(10) = 29 f(3, 10) = p(10)+p(11)+p(12)=29+31+37 = 97 f(5, 10) = p(10)+p(11)+p(12)+p(13)+p(14) = 29+31+37+41+43 = 181 f(7, 10) = p(10)+p(11)+p(12)+p(13)+p(14)+p(15)+p(16) = 29+31+37+41+43+47+53 = 281 f(9, 10) = p(10)+p(11)+p(12)+p(13)+p(14)+p(15)+p(16)+p(17)+ p(18) = 29+31+37+41+43+47+53+59+61 = 401   What is the largest value of h(n,i) that you can find?   My best solution is h(1, 7167295) = 9. Q1. What is the largest value of h(n, i) that you can find?

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