Problems & Puzzles: Puzzles

Puzzle 420. Another sequence of primes JM Bergot sent another nice puzzle He noticed that the following 5 terms are primes, starting in the prime 3: 3 3+2*5 3+2*5+4*7 3+2*5+4*7+6*11 3+2*5+4*7+6*11+8*13   The general term being: p+2*q+4*r+6*s..., where p is the starting primes & q, r, s, ... are consecutive primes p<q<r<s... In a fast search just to size the Bergot's idea I found a sequence of 8 primes with p=2344190501. Q. Can you find larger (>8) sequences of this type?

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Jacques Tramu wrote on Nov. 11 07:

Searched for all primes up to 58381861709. Found 9 sequences of length = 8 , the last one starting with p = 50166844613 . The number of sequences found is :

 ct(l) = {  2308654552, 132499192, 6146661, 640917, 31053, 1593, 142, 9, 0, ...};

(which means 9 sequences of length 8, 142 of length 7, etc.)

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J. K. Andersen wrote (Nov 15, 07)

The two first sequences of 9 primes start with p=330904463203 and
p=484653329203. Search stopped at 5*10^11.
Tramu wrote he found 9 sequences of 8 primes with start below 58381861709.
I found 11, with starting p values 2344190501, 3549549257, 3827718049,
5335587319, 7484356907, 21801443039, 27395411581, 29457402049,
30390389101, 50166844613, 58258411313.

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