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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.
4
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.
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).
Sequence in context: A160248 A356044 A317858 * A212891 A107983 A009278
KEYWORD
nonn
AUTHOR
Paolo Xausa, Aug 25 2024
STATUS
approved