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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.
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%I #18 Mar 29 2023 09:19:06

%S 1,3,5,8,10,14,15,20,21,24,26,33,27,34,35,38,41,41,33,45,37,41,46,63,

%T 36,31,31,25,35,50,39,56,23,15,25,14,-6,8,-5,-3,-6,3,-49,6,-6,-15,-8,

%U -9,-78,-124,-112,-100,-118,-122,-133,-109,-110,-139,-127,-117,-237,-166,-185,-218,-171,-215

%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 v.

%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, 2]], {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 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

%Y Cf. A345415-A345428.

%K sign

%O 1,2

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