Problems & Puzzles: Puzzles

 Puzzle 741. Find the smallest set of distinct primes such that... Q. Find the smallest set of distinct prime numbers such that in the set you have exactly one 1, two 2s, three 3s,... nine 9s. (The smallest set means here the smallest sum of the primes in the set) Note: Just in case you can not found a solution as the required one you are able to switch from "exactly" to "at least"

Contributions came from Giovanni Resta, Emmanuel Vantieghem, Claudio Meller, J. K. Andersen, Hakan Summakoğlu & Jan van Delden.

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Giovanni and Claudio Meller found both and independently, the minimal solution for the "exactly" version of this puzzle:

{5, 7, 29, 47, 59, 61, 67, 79, 83, 89, 269, 389, 449, 487, 569, 587,
659, 683, 887} minimizes the sum (5505),

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Emanuel wrote:

My best solution is:

{ 3,5,29,59,61,67,79,83,89,97,269,449,467,487,569,587,683,859,887 }  with sum  5829

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J. K. wrote:

A limited search by hand found a set with sum 5586: {5, 47, 59, 61,
67, 79, 83, 89, 269, 283, 389, 467, 487, 499, 569, 587, 659, 887}
I doubt it's the smallest.

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Hakan wrote:

My smallest sum is 6009.

Set={5, 7, 29, 47, 59, 61, 67, 79, 83, 89, 269, 463, 467, 487, 569, 599, 859, 883, 887}

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Jan wrote:

My best solution so far has sum 5685: [5, 7, 29, 47, 59, 61, 67, 79, 83, 89, 269, 389, 449, 467, 569, 587, 683, 859, 887]

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Considering sets with exactly zero 0's, one 1's, two 2's, ..., nine 9's,
the set

the set
{5, 59, 67, 83, 89, 269, 281, 389, 467, 479, 487, 499, 569, 587, 659, 683, 787} minimizes the largest value,

the set
{2,5,7,29,47,59,61,67,79,83,89,97,449,569,647,659,683,859,883,887}
maximizes the number of elements (20),

and the set
{2, 99999999988888888777777766666655555444423313} maximizes the sum.

If we are looking for sets with at least one 1's, two 2's, ..., nine 9's, then

the set {5, 7, 19, 29, 41, 43, 47, 53, 59, 61, 67, 79, 83, 89, 97, 149,
151, 157, 163, 167, 181, 269, 281, 283, 383, 389, 683} minimizes the sum (4035) and

the set {5, 19, 29, 41, 47, 53, 59, 61, 67, 79, 83, 89, 97, 149, 151,
157, 163, 167, 181, 263, 269, 281, 283, 383, 389, 487} minimizes the largest value.

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