A345427 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 v.

1, 3, 5, 8, 10, 14, 15, 20, 21, 24, 26, 33, 27, 34, 35, 38, 41, 41, 33, 45, 37, 41, 46, 63, 36, 31, 31, 25, 35, 50, 39, 56, 23, 15, 25, 14, -6, 8, -5, -3, -6, 3, -49, 6, -6, -15, -8, -9, -78, -124, -112, -100, -118, -122, -133, -109, -110, -139, -127, -117, -237, -166, -185, -218, -171, -215

Offset

1,2

Comments

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

Links

Table of n, a(n) for n=1..66.

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, 2]], {x, 1, n}, {y, 1, n}];
Table[Print[n, " ", a[n]]; a[n], {n, 1, 62}] (* Jean-François Alcover, Mar 28 2023 *)

Prog

(Python)
from sympy.core.numbers import igcdex
def A345427(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))) # Chai Wah Wu, Jun 22 2021

Crossrefs

Cf. A345415-A345428.

Sequence in context: A097911 A051611 A258028 * A005004 A006218 A062839

Adjacent sequences: A345424 A345425 A345426 * A345428 A345429 A345430

Keyword

sign

Author

N. J. A. Sloane, Jun 22 2021

Status

approved