A345423 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 u.

0, 1, 2, 3, 4, 5, 5, 7, 6, 6, 7, 9, 2, 7, 5, 3, 5, 2, -7, 1, -9, -8, -4, 4, -25, -25, -26, -40, -31, -19, -31, -17, -53, -65, -57, -71, -92, -71, -79, -91, -95, -85, -138, -88, -100, -115, -109, -125, -195, -215, -207, -191, -210, -213, -227, -199, -193, -233, -222, -238

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

Maple

mygcd:=proc(a,b) local d,s,t; d := igcdex(a,b,`s`,`t`); [a,b,d,s,t]; end;
ansu:=[]; ansv:=[]; ansb:=[];
for N from 1 to 80 do
tu:=0; tv:=0; tb:=0;
for x from 1 to N do
for y from 1 to N do
if igcd(x,y)=1 then
   tu:=tu + mygcd(x,y)[4];
   tv:=tv + mygcd(x,y)[5];
   tb:=tb + mygcd(x,y)[4] + mygcd(x,y)[5];
fi;
od: od:
ansu:=[op(ansu),tu];
ansv:=[op(ansv),tv];
ansb:=[op(ansb),tb];
od:
ansu; # the present sequence
ansv; # A345424
ansb; # A345425
# for A345426, A345427, A345428, omit the "igcd(x,y)=1" test

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, 1]], 0], {x, 1, n}, {y, 1, n}];
Table[Print[n, " ", a[n]]; a[n], {n, 1, 60}] (* Jean-François Alcover, Mar 28 2023 *)

Python

from sympy.core.numbers import igcdex
def A345423(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)) if w == 1) # Chai Wah Wu, Aug 21 2021