A294177 List of positive 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 rational, ordered by b then c.
3, 2, 1, 4, 3, 1, 5, 3, 2, 5, 4, 1, 6, 5, 1, 7, 4, 3, 7, 5, 2, 7, 6, 1, 8, 2, 1, 8, 5, 3, 8, 7, 1, 9, 3, 2, 9, 5, 4, 9, 7, 2, 9, 7, 6, 9, 8, 1, 9, 8, 6, 10, 7, 3, 10, 9, 1, 11, 6, 5, 11, 7, 4, 11, 8, 3, 11, 9, 2, 11, 10, 1, 11, 10, 8, 12, 7, 5, 12, 11, 1, 13, 7, 6, 13, 8, 5, 13, 9, 4, 13, 10, 3, 13, 11, 2, 13, 12, 1, 14, 5, 3, 14, 8, 3, 14, 8
Offset: 1
Examples
1/3^2 + 1/2^2 + 1/1^2 = 1/(6/7)^2.
Links
- David A. Corneth, Table of n, a(n) for n = 1..22698
Crossrefs
Cf. A299170.
Programs
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Mathematica
n = 16; lst = {}; Do[ Do[Do[If[GCD[b, c, d] == 1, r = Sqrt[1/(1/b^2 + 1/c^2 + 1/d^2)]; If[Element[r, Integers] || Element[r, Rationals], lst = AppendTo[lst, {b, c, d}]]], {d, c - 1}], {c, b - 1}], {b, n}]; lst//Flatten -
PARI
upto(n) = {my(res = List(), sd, stepd); for(b = 3, n, for(c = 2, b - 1, if((b - c) % 2 == 0, sd = b % 2 + 1; stepd = 2, sd = 1; stepd = 1); forstep(d = sd, c - 1, stepd, if(issquare((b*d)^2 + (b*c)^2 + (c*d)^2) && gcd([b, c, d]) == 1, listput(res, [b, c, d]))))); concat(Vec(res))} \\ David A. Corneth, Dec 29 2018
Extensions
Keyword tabf from Michel Marcus, Jan 18 2019
Comments