A195770 Positive integer a is repeated m times, where m is the number of 1-Pythagorean triples (a,b,c) satisfying a<=b.
3, 5, 6, 7, 7, 9, 9, 10, 11, 11, 12, 13, 13, 14, 14, 15, 15, 15, 16, 17, 17, 18, 18, 19, 19, 20, 21, 21, 21, 21, 22, 22, 23, 23, 24, 24, 25, 25, 25, 26, 26, 27, 27, 27, 28, 28, 29, 29, 30, 30, 30, 31, 31, 32, 32, 32, 33, 33, 33, 33
Offset: 1
Keywords
Examples
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
Links
- Robert Israel, Table of n, a(n) for n = 1..10000
Programs
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Maple
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 -
Mathematica
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 *)
Extensions
Name corrected by Robert Israel, Jul 04 2024
Comments