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.
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
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 *)
PROG
(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
CROSSREFS
KEYWORD
sign
AUTHOR
N. J. A. Sloane, Jun 22 2021
STATUS
approved