Problem 42355. Minimum Set (A+A)U(A*A) OEIS A263996

Created by Richard Zapor

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This Challenge is to find an integer vector A that creates the minimum set size for (A+A) U (A*A) for a given vector length, <a href = "https://oeis.org/A263996">OEIS A263996</a>. The length, best value, Prime_max, and Value_max will be provided.The <a href = "https://oeis.org/A263996">OEIS A263996</a> gives the minimum set sizes thru length 50. Length 7 has best value 26 with Prime_max 5 and Value_max 8. A=[1 2 3 4 5 6 8] yields [1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 18 20 24 25 30 32 36 40 48 64].
The <a href = "http://68.173.157.131/Contest/SumsAndProducts1/FinalReport">Al Zimmermann Sums Contest Final Report</a> extends A263996 for lengths 40:40:1000 with complete vector solutions. The contest winner, Rokicki, noted his method used P-smooth sets, hill climbing, and random swaps. The contest was a little tougher with only L given.Example Input/Output:
L=9;Best=36;pmax=5;vmax=12;
v = SP(L,Best,pmax,vmax); Yields v=[1 2 3 4 5 6 8 10 12]Theory/Hints: The V superset is found using <a href = "http://www.mathworks.com/matlabcentral/cody/problems/1298-p-smooth-numbers">psmooth(pmax,vmax)</a> . One observation is that for every v element the set v contains prod of all v element factor permutations. The time eater will be score evaluation. Residual evaluation suggested. A history screen, prior to score evaluation, of prior processed vectors is essential. A quick history pre-screen is vector sum. Replace testing of only values that are not factors of other numbers (eg 2,3 no replace) enables a reasonable time rolling score solution without random for the small test case values.

This Challenge is to find an integer vector A that creates the minimum set size for (A+A) U (A*A) for a given vector length, <https://oeis.org/A263996 OEIS A263996>. The length, best value, Prime_max, and Value_max will be provided.

The <https://oeis.org/A263996 OEIS A263996> gives the minimum set sizes thru length 50. Length 7 has best value 26 with Prime_max 5 and Value_max 8. A=[1 2 3 4 5 6 8] yields [1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 18 20 24 25 30 32 36 40 48 64].
The <http://68.173.157.131/Contest/SumsAndProducts1/FinalReport Al Zimmermann Sums Contest Final Report> extends A263996 for lengths 40:40:1000 with complete vector solutions. The contest winner, Rokicki, noted his method used P-smooth sets, hill climbing, and random swaps. The contest was a little tougher with only L given.

Example Input/Output:
L=9;Best=36;pmax=5;vmax=12;
v = SP(L,Best,pmax,vmax); Yields v=[1 2 3 4 5 6 8 10 12]

Theory/Hints: The V superset is found using <http://www.mathworks.com/matlabcentral/cody/problems/1298-p-smooth-numbers psmooth(pmax,vmax)> . One observation is that for every v element the set v contains prod of all v element factor permutations. The time eater will be score evaluation. Residual evaluation suggested. A history screen, prior to score evaluation, of prior processed vectors is essential. A quick history pre-screen is vector sum. Replace testing of only values that are not factors of other numbers (eg 2,3 no replace) enables a reasonable time rolling score solution without random for the small test case values.

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Problem 42355. Minimum Set (A+A)U(A*A) OEIS A263996

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