A345426 For 1<=x<=n, 1<=y<=n, write gcd(x,y) = u*x+v*y with u,v minimal; a(n) = sum of the values of u.

0, 1, 2, 4, 5, 8, 8, 12, 12, 14, 15, 21, 14, 20, 20, 22, 24, 23, 14, 25, 16, 19, 23, 39, 11, 5, 4, -3, 6, 20, 8, 24, -10, -19, -10, -22, -43, -30, -44, -43, -47, -39, -92, -38, -51, -61, -55, -57, -127, -174, -163, -152, -171, -176, -188, -165, -167, -197, -186, -177, -298, -228

Offset: 1

Author

N. J. A. Sloane, Jun 22 2021

Keywords

• sign

Comments

Minimal means minimize u^2+v^2. We follow Maple, PARI, etc., in setting u=0 and v=1 when x=y.

Crossrefs

Cf. A345415-A345428.

Programs

Mathematica

T[x_, y_] := T[x, y] = Module[{u, v}, MinimalBy[{u, v} /. Solve[u^2 + v^2 <= x^2 + y^2 && u*x + v*y == GCD[x, y], {u, v}, Integers], #.# &]];
a[n_] := a[n] = Sum[T[x, y][[1, 1]], {x, 1, n}, {y, 1, n}];
Table[Print[n, " ", a[n]]; a[n], {n, 1, 62}] (* Jean-François Alcover, Mar 28 2023 *)

Python

from sympy.core.numbers import igcdex
def A345426(n): return sum(u for u, v, w in (igcdex(x, y) for x in range(1, n+1) for y in range(1, n+1))) # Chai Wah Wu, Jul 01 2021