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A285493 a(n) is the least positive integer not already appearing such that no two distinct unordered pairs of points ((n, a(n)), (m, a(m))) and ((k, a(k)), (j, a(j))) have the same midpoint.

%I A285493 #22 Dec 12 2023 13:47:08
%S A285493 1,2,4,3,6,10,15,5,13,9,18,29,7,25,37,8,22,14,41,48,23,58,11,66,32,78,
%T A285493 24,52,83,12,73,93,26,60,42,118,21,89,65,106,139,145,19,84,16,162,76,
%U A285493 43,173,183,199,123,87,30,28,161,101,56,116,55,235,182,150
%N A285493 a(n) is the least positive integer not already appearing such that no two distinct unordered pairs of points ((n, a(n)), (m, a(m))) and ((k, a(k)), (j, a(j))) have the same midpoint.
%C A285493 No three terms a(j), a(j+k), a(j+2k) (for any j and k) form an arithmetic progression.
%C A285493 Conjecture: This is a permutation of the positive integers.
%H A285493 Giovanni Resta, <a href="/A285493/b285493.txt">Table of n, a(n) for n = 1..3000</a> (first 650 terms from Peter Kagey)
%e A285493 a(3) != 3 or else midpoint((3,3), (1,1)) = midpoint((2,2), (2,2)), thus
%e A285493 a(3) = 4.
%e A285493 a(6) != 5 or else midpoint((6,5), (3,4)) = midpoint((4,3), (5,6));
%e A285493 a(6) != 7 or else midpoint((6,7), (1,1)) = midpoint((2,2), (5,6));
%e A285493 a(6) != 8 or else midpoint((6,8), (2,2)) = midpoint((3,4), (5,6));
%e A285493 a(6) != 9 or else midpoint((6,9), (4,3)) = midpoint((5,6), (5,6)); thus
%e A285493 a(6) = 10.
%Y A285493 Cf. A094870, A285491, A285492.
%K A285493 nonn,look
%O A285493 1,2
%A A285493 _Peter Kagey_, Apr 19 2017