A299170 List of integer triples (b,c,d) where b > c > d are coprime and 1/b^2 + 1/c^2 + 1/d^2 = 1/r^2 and r is an integer, ordered by b then c.

156, 65, 45, 156, 80, 65, 255, 136, 90, 255, 160, 136, 609, 580, 315, 609, 580, 560, 1295, 444, 315, 1295, 560, 444, 1428, 221, 91, 1560, 1547, 170, 1640, 369, 270, 1640, 480, 369, 1833, 884, 799, 1924, 663, 629, 2385, 1484, 945, 2385, 1680, 1484, 2925, 1100, 429

Offset: 1

Author

Ralf Steiner, Feb 04 2018

Keywords

• nonn
• tabf

Comments

Conjectures:

12|r, 3|b or 3|c or 3|d, 4|b or 4|c or 4|d.

No term is powerful (A001694) or square (A000290).

Examples

			1/156^2 + 1/65^2 + 1/45^2 = 1/36^2 = 1/(12*3)^2.
As an array, sequence begins:
   156,   65,   45
   156,   80,   65,
   255,  136,   90,
   255,  160,  136,
   609,  580,  315,
   609,  580,  560,
  1295,  444,  315,
  1295,  560,  444,
  1428,  221,   91,
  1560, 1547,  170,
  1640,  369,  270,
  1640,  480,  369,
  1833,  884,  799,
  1924,  663,  629,
  ...
		

Links

• Giovanni Resta, Table of n, a(n) for n = 1..570

Crossrefs

Cf. A065607, A120692, A120693.

Programs

Mathematica

n = 1500; lst = {}; Do[Do[Do[If[GCD[b, c, d] == 1,
r = Sqrt[1/(1/b^2 + 1/c^2 + 1/d^2)];
  If[IntegerQ[r], lst = AppendTo[lst, {b, c, d}]]], {d, c - 1}],
{c, b - 1}], {b, n}]; lst//Flatten

Formula

a(n) > 1.

Extensions

a(28)-a(51) from Giovanni Resta, Feb 06 2018