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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.
2
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
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
KEYWORD
nonn,tabf
AUTHOR
Ralf Steiner, Feb 04 2018
EXTENSIONS
a(28)-a(51) from Giovanni Resta, Feb 06 2018
STATUS
approved