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

1,1

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).

Links

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

Formula

a(n) > 1.

Example

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,
  ...

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

Crossrefs

Cf. A065607, A120692, A120693.

Sequence in context: A115466 A057966 A247435 * A112818 A047635 A295221

Adjacent sequences: A299167 A299168 A299169 * A299171 A299172 A299173

Keyword

nonn,tabf

Author

Ralf Steiner, Feb 04 2018

Extensions

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

Status

approved