A338886 -id:A338886 - cp's OEIS frontend

A338885 Irregular triangle read by rows in which the n-th row lists all numbers k such that there exists a diagonal lattice rectangle touching all four sides of an n X k rectangle.

2, 3, 4, 5, 4, 5, 7, 6, 9, 10, 5, 7, 8, 11, 13, 7, 8, 10, 13, 16, 17, 6, 9, 11, 12, 15, 19, 21, 6, 8, 10, 11, 14, 17, 22, 25, 26, 7, 9, 10, 11, 13, 14, 16, 17, 19, 25, 29, 31, 9, 12, 13, 15, 18, 20, 21, 28, 33, 36, 37, 7, 8, 11, 12, 13, 14, 15, 17, 20, 22, 23

Offset: 2

Peter Kagey, Nov 14 2020

A diagonal lattice rectangle is a rectangle with integer coordinates and no side parallel to the x-axis.
Conjecture: The smallest number in the n-th row is A228286(n).
Conjecture: The largest number in the n-th row is A033638(n).

			Table begins:
   n | n-th row
-----+------------------------------------------------
   2 | 2
   3 | 3
   4 | 4,  5
   5 | 4,  5,  7
   6 | 6,  9, 10
   7 | 5,  7,  8, 11, 13
   8 | 7,  8, 10, 13, 16, 17
   9 | 6,  9, 11, 12, 15, 19, 21
  10 | 6,  8, 10, 11, 14, 17, 22, 25, 26
  11 | 7,  9, 10, 11, 13, 14, 16, 17, 19, 25, 29, 31
  12 | 9, 12, 13, 15, 18, 20, 21, 28, 33, 36, 37
For n = 6, three of the diagonal lattice rectangles that touch the y-axis, x-axis, and line x = 6 are:
(2 ,6), (0,2), (4,0), (6,4);
(2, 9), (0,8), (4,0), (6,1); and
(3,10), (0,9), (3,0), (6,1);
which have maximum y-values of 6, 9, and 10 respectively.

Peter Kagey, Table of n, a(n) for n = 2..11808 (first 100 rows, flattened)
Code Golf Stack Exchange, Rectangles in rectangles

Cf. A033638, A085582, A113751, A228286.

Cf. A338886 (row lengths).

A338887 a(n) is the size of the set {x + yz | xy + z = n}, where x, y, and z are positive integers.

Original entry on oeis.org

0, 1, 1, 2, 3, 3, 5, 6, 7, 9, 12, 10, 14, 15, 16, 19, 23, 19, 27, 24, 28, 30, 34, 30, 42, 38, 39, 39, 48, 42, 58, 50, 53, 56, 65, 51, 71, 67, 67, 68, 85, 68, 88, 76, 82, 85, 98, 76, 107, 95, 101, 94, 117, 96, 118, 108, 114, 121, 137, 100, 142, 131, 134, 130, 154

Offset: 1

Peter Kagey, Nov 14 2020

nonn

			The a(7) = 5 solutions are:
2 + 3 * 1 =  5 (corresponding to 2 * 3 + 1 = 7),
1 + 6 * 1 =  7 (corresponding to 1 * 6 + 1 = 7),
2 + 2 * 3 =  8 (corresponding to 2 * 2 + 3 = 7),
1 + 5 * 2 = 11 (corresponding to 1 * 5 + 2 = 7), and
1 + 4 * 3 = 13 (corresponding to 1 * 4 + 3 = 7).

Peter Kagey, Table of n, a(n) for n = 1..1000

Cf. A228286, A338886.

a(n) <= A338886(n).

cp's OEIS Frontend

A338885 Irregular triangle read by rows in which the n-th row lists all numbers k such that there exists a diagonal lattice rectangle touching all four sides of an n X k rectangle.

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A338887 a(n) is the size of the set {x + yz | xy + z = n}, where x, y, and z are positive integers.

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cp's OEIS Frontend

A338885 Irregular triangle read by rows in which the n-th row lists all numbers k such that there exists a diagonal lattice rectangle touching all four sides of an n X k rectangle.

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A338887 a(n) is the size of the set {x + y*z | x*y + z = n}, where x, y, and z are positive integers.

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A338887 a(n) is the size of the set {x + yz | xy + z = n}, where x, y, and z are positive integers.