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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.
3
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
1,3
COMMENTS
Minimal means minimize u^2+v^2. We follow Maple, PARI, etc., in setting u=0 and v=1 when x=y.
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 *)
PROG
(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
CROSSREFS
KEYWORD
sign
AUTHOR
N. J. A. Sloane, Jun 22 2021
STATUS
approved