A285491 -id:A285491 - cp's OEIS frontend

A285490 Lexicographically earliest sequence such that no two distinct, unordered pairs of distinct points ((n, a(n)), (m, a(m))) and ((k, a(k)), (j, a(j))) have the same midpoint.

1, 1, 1, 2, 1, 3, 5, 1, 7, 4, 9, 12, 1, 17, 21, 28, 6, 27, 2, 16, 14, 22, 41, 1, 50, 11, 33, 61, 9, 4, 73, 58, 88, 97, 40, 2, 26, 19, 95, 82, 45, 15, 1, 96, 54, 66, 125, 143, 57, 138, 10, 1, 171, 92, 184, 178, 191, 54, 110, 160, 13, 3, 202, 217, 122, 88, 42

Offset: 1

Peter Kagey, Apr 19 2017

			a(1) = 1;
a(2) = 1;
a(3) = 1;
a(4) != 1 or else midpoint((1, 1), (4, 1)) = midpoint((2, 1), (3, 1));
a(4) = 2;
a(5) = 1;
a(6) != 1 or else midpoint((1, 1), (6, 1)) = midpoint((2, 1), (5, 1));
a(6) != 2 or else midpoint((3, 1), (6, 2)) = midpoint((4, 2), (5, 1));
a(6) = 3.

Giovanni Resta, Table of n, a(n) for n = 1..4000 (first 650 terms from Peter Kagey)

Cf. A285491 (pairs of not necessarily distinct points).

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.

Original entry on oeis.org

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, 24, 52, 83, 12, 73, 93, 26, 60, 42, 118, 21, 89, 65, 106, 139, 145, 19, 84, 16, 162, 76, 43, 173, 183, 199, 123, 87, 30, 28, 161, 101, 56, 116, 55, 235, 182, 150

Offset: 1

Peter Kagey, Apr 19 2017

No three terms a(j), a(j+k), a(j+2k) (for any j and k) form an arithmetic progression.

Conjecture: This is a permutation of the positive integers.

			a(3) != 3 or else midpoint((3,3), (1,1)) = midpoint((2,2), (2,2)), thus
a(3) = 4.
a(6) != 5 or else midpoint((6,5), (3,4)) = midpoint((4,3), (5,6));
a(6) != 7 or else midpoint((6,7), (1,1)) = midpoint((2,2), (5,6));
a(6) != 8 or else midpoint((6,8), (2,2)) = midpoint((3,4), (5,6));
a(6) != 9 or else midpoint((6,9), (4,3)) = midpoint((5,6), (5,6)); thus
a(6) = 10.

Giovanni Resta, Table of n, a(n) for n = 1..3000 (first 650 terms from Peter Kagey)

Cf. A094870, A285491, A285492.

A319479 Lexicographically earliest sequence such that for every n and every sequence 1 <= b_1 < b_2 < ... < b_t = n, the values of barycenter((b_1, a(b_1)), (b_2, a(b_2)), ..., (b_t, a(b_t))) are distinct.

Original entry on oeis.org

1, 1, 2, 1, 4, 8, 16, 29, 51, 128, 344, 528, 1863, 5283, 12445

Offset: 1

Peter Kagey, Sep 19 2018

The barycenter of the points {(x_1, y_1), (x_2, y_2), ..., (x_k, y_k)} is given by the average of the x and y coordinates: (Sum_{i=1..k} x_i/k, Sum_{i=1..k} y_i/k).

			For n = 5, a(5) = 4 because letting a(5) = 1, 2, or 3 creates barycenter collisions:
+-----------+----------------+--------------------------------+------------+
| candidate | set 1          | set 2                          | barycenter |
+-----------+----------------+--------------------------------+------------+
| (5, 1)    | (1, 1); (5, 1) | (1, 1); (2, 1); (4, 1); (5, 1) | (6, 1)     |
| (5, 2)    | (2, 1); (5, 2) | (2, 1); (3, 2); (4, 1); (5, 2) | (3.5, 1.5) |
| (5, 3)    | (1, 1); (5, 3) | (1, 1); (3, 2); (5, 3)         | (3, 2)     |
+-----------+----------------+--------------------------------+------------+
And a(5) = 4 creates no such problems.

Cf. A285491.

cp's OEIS Frontend

A285490 Lexicographically earliest sequence such that no two distinct, unordered pairs of distinct points ((n, a(n)), (m, a(m))) and ((k, a(k)), (j, a(j))) have the same midpoint.

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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.

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A319479 Lexicographically earliest sequence such that for every n and every sequence 1 <= b_1 < b_2 < ... < b_t = n, the values of barycenter((b_1, a(b_1)), (b_2, a(b_2)), ..., (b_t, a(b_t))) are distinct.

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