Problems & Puzzles: Puzzles

Puzzle 876.Prime Tuples

Dmitry Kamenetsky very recently, published a paper related to "a variety of prime-generating constructions that are based on sums of primes".

Let's continue with a second issue of his paper, its third issue, the "Prime Tuples".

Definitions:

"A prime tuple of order n (odd) with length k is an array of distinct odd primes (p0, p1, . . . , pk−1), such that every term after the n-th term is the sum of the previous n terms"

"The weight of a prime tuple of order n is the sum of its first n terms."

"When two tuples of the same order have the same length, then we prefer the one with the smaller weight."

k is the total of primes of the set, before a composite emerges applying recursively the rule.

Here are the best results gotten by Dmitry, according to his Table 5:

 Order n Prime set with only the first n terms Length k Weigth, w Primes generated, k-n 3 3, 13, 7 7 23 4 5 17, 3, 19, 7, 13 11 59 6 5 17, 5, 11, 23, 3 11 59 6 7 157, 379, 487, 109, 13, 7, 271 25 1423 18 9 11, 47, 17, 23, 41, 5, 3, 13, 19 19 179 10 11 43, 7, 19, 13, 3, 17, 11, 5, 29, 41, 23 23 211 12 13 53, 137, 11, 17, 41, 227, 47, 101, 83, 5, 149, 263, 29 34 1163 21 15 29, 5, 23, 11, 41, 47, 89, 17, 71, 3, 7, 13, 37, 19, 79 31 491 16 17 5, 47, 53, 11, 17, 41, 89, 3, 61, 43, 97, 19, 13, 7, 37, 31, 73 35 647 18 19 89, 227, 29, 17, 5, 251, 269, 107, 101, 197, 41, 191, 173, 179, 47, 53, 71, 11, 23 43 2081 24 Example: 3, 13, 7, 23, 43, 73, 139, 255 (composite) in black the first n prime terms, in blue the primes generated, in red the composite not a member of the set.

Q1.Can you improve some of the results shown in the table above?.

Q2. Can you extend his Table with your best results?

Q3. Now, let's forget the weight, just for each odd n=>3 find the first n primes that produce a maximal k.

Contributions came from Claudio Meller.

***

Claudio made two observations related to Q3, but limited only to the case n=3

a) Considering just the rightmost digit of the sequence of primes produced, the largest quantity of primes produced before a digit "5" appears as the rightmost digit is 11, so kmax=14.

b) But considering just the divisibility by 3 of the members of the sequence produced the largest sequence of primes produced is 3, so kmax=6, except that one of the three initial primes is the prime 3, then kmax=7.

I [CR] wonder what could be the results of n>3 following the approach of Claudio. Perhaps this issue could be the matter of the a future puzzle? I will consider it...

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