This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
The first seven 1-Pythagorean triples (a,b,c), ordered as described above, are as follows: 3,5,7........7^2 = 3^2 + 5^2 + 3*5 5,16,19.....19^2 = 5^2 + 16^2 + 5*16 6,10,14.....14^2 = 6^2 + 10^2 + 6*10 7,8,13 7,33,37 9,15,21 9,56,61 10,32,38
f:= proc(a) local F,r,u,b;
r:= 3*a^2;
nops(select(proc(t) local b; b:= (r/t - t - 2*a)/4;
(t + r/t) mod 4 = 0 and b::integer and b >= a end proc, numtheory:-divisors(3*a^2)));
end proc:
seq(a$f(a),a=1..100); # Robert Israel, Jul 04 2024
z8 = 2000; z9 = 400; z7 = 100;
k = 1; c[a_, b_] := Sqrt[a^2 + b^2 + k*a*b];
d[a_, b_] := If[IntegerQ[c[a, b]], {a, b, c[a, b]}, 0]
t[a_] := Table[d[a, b], {b, a, z8}]
u[n_] := Delete[t[n], Position[t[n], 0]]
Table[u[n], {n, 1, 15}]
t = Table[u[n], {n, 1, z8}];
Flatten[Position[t, {}]]
u = Flatten[Delete[t, Position[t, {}]]];
x[n_] := u[[3 n - 2]];
Table[x[n], {n, 1, z7}] (* this sequence *)
y[n_] := u[[3 n - 1]];
Table[y[n], {n, 1, z7}] (* A195866 *)
z[n_] := u[[3 n]];
Table[z[n], {n, 1, z7}] (* A195867 *)
x1[n_] := If[GCD[x[n], y[n], z[n]] == 1, x[n], 0]
y1[n_] := If[GCD[x[n], y[n], z[n]] == 1, y[n], 0]
z1[n_] := If[GCD[x[n], y[n], z[n]] == 1, z[n], 0]
f = Table[x1[n], {n, 1, z9}];
x2 = Delete[f, Position[f, 0]] (* A195868 *)
g = Table[y1[n], {n, 1, z9}];
y2 = Delete[g, Position[g, 0]] (* A195869 *)
h = Table[z1[n], {n, 1, z9}];
z2 = Delete[h, Position[h, 0]] (* A195870 *)
As the first four primitive Pythagorean triples (ordered by increasing A) are (3,4,5), (5,12,13), (7,24,25) and (8,15,17), then a(1)=4, a(2)=12, a(3)=24 and a(4)=15.
a156678 n = a156678_list !! (n-1)
a156678_list = f 1 1 where
f u v | v > uu `div` 2 = f (u + 1) (u + 2)
| gcd u v > 1 || w == 0 = f u (v + 2)
| otherwise = v : f u (v + 2)
where uu = u ^ 2; w = a037213 (uu + v ^ 2)
-- Reinhard Zumkeller, Nov 09 2012
PrimitivePythagoreanTriplets[n_]:=Module[{t={{3,4,5}},i=4,j=5},While[i
As the first four Pythagorean triples (ordered by increasing A) are (3,4,5), (5,12,13), (6,8,10) and (7,24,25), then a(1)=5, a(2)=13, a(3)=10 and a(4)=25.
PythagoreanTriplets[n_]:=Module[{t={{3,4,5}},i=4,j=5},While[i
2*3 + 2*3 = 3*4 3*4 + 5*6 = 6*7 4*5 + 9*10 = 10*11 5*6 + 6*7 = 8*9 5*6 + 14*15 = 15*16 6*7 + 9*10 = 11*12
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