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 A193838 Size k of smallest square of k X k lattice points from which n points with distinct mutual distances can be chosen. 7
 1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 13, 15, 16 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 REFERENCES R. K. Guy, Unsolved Problems in Number Theory, Third Edition, Springer New York, 2004, F2, 367-368. Keith F. Lynch, Posting to Math Fun Mailing List, Apr 02 2016. LINKS P. ErdÅ‘s and R. K. Guy, Distinct distances between lattice points, Elemente der Mathematik 25 (1970), 121-123. Sean A. Irvine, Java program (github) Samuel B. Reid, The unique solution that causes a(7) to be 7. Wolfram Demonstration Project, No Repeated Distances. A. Zimmermann, Al Zimmermann's Programming Contests: Point Packing. (Oct 10, 2009). EXAMPLE a(1) is the degenerate case of a single point, a(2)=2 is trivial. a(3)=3: The points ((1,2),(3,1),(3,2)) have distinct mutual squared distances 1, 4, 5. a(8)=9 is the first square for which k>n: ((1,1), (1,4), (2,2), (6,1), (7,6), (7,7), (9,2), (9,4)) have 7*8/2=28 mutual squared distances: 1, 2, 4, 5, 8, 9, 10, 13, 17, 18, 20, 25, 26, 29, 34, 37, 40, 41, 45, 49, 50, 53, 61, 64, 65, 68, 72, 73, and no configuration of 8 points fitting on an 8 X 8 square exists. a(10)=11, only two subsets barring symmetry:   {(0,0), (0,2), (0,3), (0,7), (1,10), (5,4), (6,0), (8,7), (9,8), (10, 10)},   {(0,0), (0,6), (0,7), (1,2), (4,10), (7,8), (7,10), (9,2), (9,6), (10,5)}. a(11)=13, one of the four subsets of the 12 X 13 grid, barring symmetry: {(0,0), (0,1), (0,9), (0,12), (2,0), (5,3), (6,12), (7,0), (8,4), (10,10), (11,11)} a(12)=15 is satisfied by {(0,0), (1,0), (1,12), (3,0), (7,0), (7,14), (9,4), (12,11), (13,3), (13,8), (14,2), (14,13)}. - Sean A. Irvine, Jul 13 2020 a(13)=16 is satisfied by {(1,1), (2,2), (2,16), (4,14), (6,14), (7,16), (8,8), (11,2), (11,5), (13,15), (13,16), (16,1), (16,8)}. - Bert Dobbelaere, Sep 20 2020 CROSSREFS Cf. A193839, A003022. See A271490 for the inverse function. Sequence in context: A062462 A033948 A285514 * A117730 A174328 A272570 Adjacent sequences:  A193835 A193836 A193837 * A193839 A193840 A193841 KEYWORD nonn,hard,more AUTHOR Hugo Pfoertner, Aug 06 2011 EXTENSIONS a(10)-a(11) corrected by Ehit Dinesh Agarwal, May 28 2020 a(12) from Sean A. Irvine, Jul 13 2020 a(13) from Bert Dobbelaere, Sep 20 2020 STATUS approved

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