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

3,1

Comments

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

Links

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

Formula

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

a(n) = (A375676(n) - 2*n)/2.

a(n) = n for n = 4*k (k >= 1).

Example

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.

Mathematica

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]

Crossrefs

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

Cf. A262767, A329402, A306841.

Sequence in context: A160248 A356044 A317858 * A212891 A107983 A009278

Adjacent sequences: A375670 A375671 A375672 * A375674 A375675 A375676

Keyword

nonn

Author

Paolo Xausa, Aug 25 2024

Status

approved