A345424 For 1<=x<=n, 1<=y<=n, with gcd(x,y)=1, write 1 = gcd(x,y) = u*x+v*y with u,v minimal; a(n) = sum of the values of v.

1, 2, 3, 4, 5, 6, 6, 8, 7, 7, 8, 10, 3, 8, 6, 4, 6, 3, -6, 2, -8, -7, -3, 5, -24, -24, -25, -39, -30, -18, -30, -16, -52, -64, -56, -70, -91, -70, -78, -90, -94, -84, -137, -87, -99, -114, -108, -124, -194, -214, -206, -190, -209, -212, -226, -198, -192, -232, -221, -237, -358, -277, -287, -337

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 == 1, {u, v}, Integers], #.# &]];
a[n_] := a[n] = Sum[If[GCD[x, y] == 1, T[x, y][[1, 2]], 0], {x, 1, n}, {y, 1, n}];
Table[Print[n, " ", a[n]]; a[n], {n, 1, 64}] (* Jean-François Alcover, Mar 28 2023 *)

Python

from sympy.core.numbers import igcdex
def A345424(n): return sum(v for u, v, w in (igcdex(x,y) for x in range(1,n+1) for y in range(1,n+1)) if w == 1) # Chai Wah Wu, Jun 22 2021