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

5, 17, 41, 41, 47, 127, 77, 81, 171, 132, 99, 283, 94, 205, 349, 158, 115, 457, 122, 296, 530, 267, 134, 546, 219, 260, 428, 471, 130, 953, 144, 264, 613, 332, 557, 1031, 139, 346, 614, 600, 162, 1381, 169, 562, 1132, 348, 186, 1000, 363, 593, 688, 571, 164, 1123

Offset: 1

Author

Jeppe Stig Nielsen, Feb 19 2020

Keywords

• nonn

Comments

Gives the row lengths of the irregular array A290451.

Examples

			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.
		

Links

• James Grime and Brady Haran, Superhero Triangles, Numberphile video (2020).

Crossrefs

Cf. A007237, A290451.

Programs

Mathematica

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 *)

Python

from math import sqrt
def A332689(n):
    L = []; k = 4*n*n
    for x in range(1, int(2*sqrt(3)*n) + 1):
        for y in range(max(int(k/x) + 1, x), int((k + 2*n*sqrt(k + x*x))/x) + 1):
            if k*(x+y)%(x*y-k) == 0:
                s = x + y + k*(x+y)//(x*y-k)
                if s not in L: L.append(s)
    return len(L)  # Ya-Ping Lu, Dec 28 2023

Extensions

a(8)-a(54) from Giovanni Resta, Mar 04 2020