cp's OEIS Frontend

A332689 Number of distinct areas of integer-sided triangles whose area equals n times their perimeter.

%I A332689 #27 Dec 29 2023 08:06:21
%S A332689 5,17,41,41,47,127,77,81,171,132,99,283,94,205,349,158,115,457,122,
%T A332689 296,530,267,134,546,219,260,428,471,130,953,144,264,613,332,557,1031,
%U A332689 139,346,614,600,162,1381,169,562,1132,348,186,1000,363,593,688,571,164,1123
%N A332689 Number of distinct areas of integer-sided triangles whose area equals n times their perimeter.
%C A332689 Gives the row lengths of the irregular array A290451.
%H A332689 James Grime and Brady Haran, <a href="https://www.youtube.com/watch?v=UIjeCKPHbso">Superhero Triangles</a>, Numberphile video (2020).
%e A332689 For n = 2, there are 18 different (noncongruent) Heronian triangles whose area equals twice their perimeter, so A007237(2) = 18. However, two of those 18 triangles share the area 168. So there are only 17 distinct areas. Therefore, a(2) = 17.
%t A332689 a[k_] := Block[{v={},r,s,t}, Do[ If[r <= s && 4 k^2 < r s <= 12 k^2 && IntegerQ[t = 4 k^2 (r + s)/(r s - 4 k^2)] && t >= s, AppendTo[v, r + s + t]], {r, Floor[2 Sqrt[3] k]}, {s, Floor[4 k^2/r], Ceiling[12 k^2/r]}]; Length@ Union@ v]; Array[a, 20] (* _Giovanni Resta_, Mar 04 2020 *)
%o A332689 (Python)
%o A332689 from math import sqrt
%o A332689 def A332689(n):
%o A332689     L = []; k = 4*n*n
%o A332689     for x in range(1, int(2*sqrt(3)*n) + 1):
%o A332689         for y in range(max(int(k/x) + 1, x), int((k + 2*n*sqrt(k + x*x))/x) + 1):
%o A332689             if k*(x+y)%(x*y-k) == 0:
%o A332689                 s = x + y + k*(x+y)//(x*y-k)
%o A332689                 if s not in L: L.append(s)
%o A332689     return len(L)  # _Ya-Ping Lu_, Dec 28 2023
%Y A332689 Cf. A007237, A290451.
%K A332689 nonn
%O A332689 1,1
%A A332689 _Jeppe Stig Nielsen_, Feb 19 2020
%E A332689 a(8)-a(54) from _Giovanni Resta_, Mar 04 2020