A345426 - OEIS

%I #16 Mar 29 2023 09:19:00

%S 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,

%T 5,4,-3,6,20,8,24,-10,-19,-10,-22,-43,-30,-44,-43,-47,-39,-92,-38,-51,

%U -61,-55,-57,-127,-174,-163,-152,-171,-176,-188,-165,-167,-197,-186,-177,-298,-228

%N 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.

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

%t 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], #.# &]];

%t a[n_] := a[n] = Sum[T[x, y][[1, 1]], {x, 1, n}, {y, 1, n}];

%t Table[Print[n, " ", a[n]]; a[n], {n, 1, 62}] (* _Jean-François Alcover_, Mar 28 2023 *)

%o (Python)

%o from sympy.core.numbers import igcdex

%o 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

%Y Cf. A345415-A345428.

%K sign

%O 1,3

%A _N. J. A. Sloane_, Jun 22 2021