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.

%I A375673 #22 Aug 26 2024 06:05:14
%S A375673 6,4,20,12,42,8,18,15,110,12,156,35,30,16,272,36,342,20,28,99,506,24,
%T A375673 100,143,54,28,812,45,930,32,66,255,140,36,1332,323,78,40,1640,56,
%U A375673 1806,44,90,483,2162,48,294,75,102,52,2756,108,66,56,114,783,3422,60,3660,899
%N 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.
%C A375673 No such rectangle exists for n = 1 or n = 2.
%H A375673 Paolo Xausa, <a href="/A375673/b375673.txt">Table of n, a(n) for n = 3..1000</a>
%F A375673 a(n) = A375675(n)/n.
%F A375673 a(n) = (A375676(n) - 2*n)/2.
%F A375673 a(n) = n for n = 4*k (k >= 1).
%e A375673 The first rectangles are listed below.
%e A375673 .
%e A375673      |           | area/per. |    area   | perimeter
%e A375673    n |    a(n)   | (A375674) | (A375675) | (A375676)
%e A375673   ---------------------------------------------------
%e A375673    3 |      6    |     1     |     18    |     18
%e A375673    4 |      4    |     1     |     16    |     16
%e A375673    5 |     20    |     2     |    100    |     50
%e A375673    6 |     12    |     2     |     72    |     36
%e A375673    7 |     42    |     3     |    294    |     98
%e A375673    8 |      8    |     2     |     64    |     32
%e A375673    9 |     18    |     3     |    162    |     54
%e A375673   10 |     15    |     3     |    150    |     50
%e A375673   ...
%e A375673 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.
%t A375673 A375673[n_] := Module[{b, r}, SolveValues[2*r == n*b/(n+b) && b >= n, {b, r}, Integers, MaxRoots -> 1][[1,1]]];
%t A375673 Array[A375673, 100, 3]
%Y A375673 Cf. A375674 (area/perimeter), A375675 (area), A375676 (perimeter).
%Y A375673 Cf. A262767, A329402, A306841.
%K A375673 nonn
%O A375673 3,1
%A A375673 _Paolo Xausa_, Aug 25 2024