A306841 -id:A306841 - cp's OEIS frontend

A375673 n and a(n) (with a(n) >= n) are the edges of the minimum-area rectangle such that its area is an integer multiple of its perimeter.

6, 4, 20, 12, 42, 8, 18, 15, 110, 12, 156, 35, 30, 16, 272, 36, 342, 20, 28, 99, 506, 24, 100, 143, 54, 28, 812, 45, 930, 32, 66, 255, 140, 36, 1332, 323, 78, 40, 1640, 56, 1806, 44, 90, 483, 2162, 48, 294, 75, 102, 52, 2756, 108, 66, 56, 114, 783, 3422, 60, 3660, 899

Offset: 3

Paolo Xausa, Aug 25 2024

nonn

No such rectangle exists for n = 1 or n = 2.

			The first rectangles are listed below.
.
     |           | area/per. |    area   | perimeter
   n |    a(n)   | (A375674) | (A375675) | (A375676)
  ---------------------------------------------------
   3 |      6    |     1     |     18    |     18
   4 |      4    |     1     |     16    |     16
   5 |     20    |     2     |    100    |     50
   6 |     12    |     2     |     72    |     36
   7 |     42    |     3     |    294    |     98
   8 |      8    |     2     |     64    |     32
   9 |     18    |     3     |    162    |     54
  10 |     15    |     3     |    150    |     50
  ...
For n = 9, two rectangles exist with the area being an integer multiple of the perimeter: one with sides (9, 18) and one with sides (9, 74). a(9) is the smaller one.

Paolo Xausa, Table of n, a(n) for n = 3..1000

Cf. A375674 (area/perimeter), A375675 (area), A375676 (perimeter).

Cf. A262767, A329402, A306841.

A375673[n_] := Module[{b, r}, SolveValues[2*r == n*b/(n+b) && b >= n, {b, r}, Integers, MaxRoots -> 1][[1,1]]];
Array[A375673, 100, 3]

a(n) = A375675(n)/n.
a(n) = (A375676(n) - 2*n)/2.
a(n) = n for n = 4*k (k >= 1).

A375674 a(n) is the area/perimeter ratio of the rectangle whose edges are n and A375673(n).

Original entry on oeis.org

1, 1, 2, 2, 3, 2, 3, 3, 5, 3, 6, 5, 5, 4, 8, 6, 9, 5, 6, 9, 11, 6, 10, 11, 9, 7, 14, 9, 15, 8, 11, 15, 14, 9, 18, 17, 13, 10, 20, 12, 21, 11, 15, 21, 23, 12, 21, 15, 17, 13, 26, 18, 15, 14, 19, 27, 29, 15, 30, 29, 18, 16, 20, 22, 33, 17, 23, 21, 35, 18, 36, 35, 25, 19

Offset: 3

Paolo Xausa, Aug 25 2024

nonn

Paolo Xausa, Table of n, a(n) for n = 3..1000

Cf. A375673 (edges), A375675 (area), A375676 (perimeter).

Cf. A262767, A329402, A306841.

A375674[n_] := Module[{b, r}, SolveValues[2*r == n*b/(n+b) && b >= n, {b, r}, Integers, MaxRoots -> 1][[1,2]]];
Array[A375674, 100, 3]

a(n) = A375675(n)/A375676(n).

A375675 a(n) is the area of the rectangle whose edges are n and A375673(n).

Original entry on oeis.org

18, 16, 100, 72, 294, 64, 162, 150, 1210, 144, 2028, 490, 450, 256, 4624, 648, 6498, 400, 588, 2178, 11638, 576, 2500, 3718, 1458, 784, 23548, 1350, 28830, 1024, 2178, 8670, 4900, 1296, 49284, 12274, 3042, 1600, 67240, 2352, 77658, 1936, 4050, 22218, 101614, 2304, 14406

Offset: 3

Paolo Xausa, Aug 25 2024

nonn

Paolo Xausa, Table of n, a(n) for n = 3..1000

Cf. A375673 (edges), A375674 (area/perimeter), A375676 (perimeter).

Cf. A262767, A329402, A306841.

A375675[n_] := Module[{b, r}, SolveValues[2*r == n*b/(n+b) && b >= n, {b, r}, Integers, MaxRoots -> 1][[1,1]]*n];
Array[A375675, 100, 3]

a(n) = n*A375673(n).

a(n) = A375674(n)*A375676(n).

A375676 a(n) is the perimeter of the rectangle whose edges are n and A375673(n).

Original entry on oeis.org

18, 16, 50, 36, 98, 32, 54, 50, 242, 48, 338, 98, 90, 64, 578, 108, 722, 80, 98, 242, 1058, 96, 250, 338, 162, 112, 1682, 150, 1922, 128, 198, 578, 350, 144, 2738, 722, 234, 160, 3362, 196, 3698, 176, 270, 1058, 4418, 192, 686, 250, 306, 208, 5618, 324, 242, 224, 342

Offset: 3

Paolo Xausa, Aug 25 2024

nonn

Paolo Xausa, Table of n, a(n) for n = 3..1000

Cf. A375673 (edges), A375674 (area/perimeter) A375675 (area).

Cf. A262767, A329402, A306841.

A375676[n_] := Module[{b, r}, 2*(SolveValues[2*r == n*b/(n+b) && b >= n, {b, r}, Integers, MaxRoots -> 1][[1,1]] + n)];
Array[A375676, 100, 3]

a(n) = 2*(n + A375673(n)).

a(n) = A375675(n)/A375674(n).

cp's OEIS Frontend

A375673 n and a(n) (with a(n) >= n) are the edges of the minimum-area rectangle such that its area is an integer multiple of its perimeter.

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A375674 a(n) is the area/perimeter ratio of the rectangle whose edges are n and A375673(n).

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A375675 a(n) is the area of the rectangle whose edges are n and A375673(n).

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A375676 a(n) is the perimeter of the rectangle whose edges are n and A375673(n).

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