Problems & Puzzles: Puzzles

Puzzle 1018. Follow-up to Puzzle 1016

Carlos Rivera propose the following puzzle:

Redo Puzzle 1016 but relaxing the following conditions for the vector of primes.

Instead demanding the the K primes of the vector must be "ordered distinct primes", now redo the exercise accepting primes in any order and not necessarily distinct. Mwe conserve only the demand about minimal sum of these primes.

Here is what I could find.

 K Vector Sum 2 3, 7 10 3 3, 3, 7 13 4 3, 11, 7, 13 34 5 3, 11, 3, 23, 31 71 6 3, 7, 19, 11, 13, 11 64 7 3, 7, 7, 19, 23, 7, 13 79 8 3, 3, 7, 7, 3, 3, 53, 19 98

Q1. Extend the Table for K=9 to ...

On the week 20-25 Set, 2020, contributions came from:

Vicente F. Izquierdo, Oscar Volpatti, Giorgos Kalogeropoulos, Gennady Gusev and Emmanuel Vantieghem

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Vicente wrote:

The minimum value for 9 primes is 181: {3, 59, 7, 11, 7, 7, 7, 7, 73}

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Oscar wrote:

I found solutions for K = 9 to 11:

K = 9,    S = 181,  V = [3,59,7,11,7,7,7,7,73];
K = 10,  S = 184,  V = [13,11,3,7,3,37,37,7,19,47];
K = 11,  S = 215,  V = [3,3,19,3,13,37,17,11,47,3,59].

I also checked my program against your published solutions, verifying that all sums are minimal.

Only for K = 5 there's one more minimal solution:

K = 5,  S = 71,  V = [17,17,7,7,23].

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Giorgos wrote:

The new conditions didn't help me find a smaller sum...
So, I'll stick to my previous answer which is also valid for this puzzle

K=9
Vector={3,11,17,23,29,41,47,61,211}
Sum=443

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I found the second solution for K=5 with the same minimal sum =71:
7.7.23.17.17.

And two solutions for k=9 & 10 (I think not minimal)
K=9: 3.3.13.83.71.67.11.7, sum=301
K=10: 3.3.3.3.3.17.17.191.193.109, sum=542

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Emmanuel wrote:

I checked your results and agree with them.

For  K = 9  I found the vector
{3, 13, 109, 13, 19, 13, 23, 19, 23}   with sum  235.

I think it's minimal but I'm not  100%  sure.

I found a solution for  K = 10 :

{3,269,13,13,19,29,7,3,13,29}

with sum 398. But I don' think it is minimal.

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