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A345429 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|.

%I A345429 #17 Jul 13 2021 14:48:05
%S A345429 0,1,4,7,16,19,37,49,70,82,127,145,208,235,277,325,433,472,607,667,
%T A345429 757,832,1030,1102,1291,1399,1582,1708,2023,2119,2479,2671,2911,3103,
%U A345429 3409,3571,4084,4327,4669,4909,5539,5737,6430,6760,7162,7525,8353,8641,9415,9787
%N A345429 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|.
%C A345429 Minimal means minimize u^2+v^2. We follow Maple, PARI, etc., in setting u=0 and v=1 when x=y.
%C A345429 It would be nice to have b-files for this and related sequences (as listed in cross-references). The present sequence is especially interesting. What is its rate of growth?
%p A345429 mygcd:=proc(a,b) local d,s,t; d := igcdex(a,b,`s`,`t`); [a,b,d,s,t]; end;
%p A345429 ansu:=[]; ansv:=[]; ansb:=[];
%p A345429 for N from 1 to 80 do
%p A345429 tu:=0; tv:=0; tb:=0;
%p A345429 for x from 1 to N do
%p A345429 for y from 1 to N do
%p A345429 if igcd(x,y)=1 then
%p A345429    tu:=tu+abs(mygcd(x,y)[4]);
%p A345429    tv:=tv+abs(mygcd(x,y)[5]);
%p A345429    tb:=tb+mygcd(x,y)[4]^2 + mygcd(x,y)[5]^2;
%p A345429 fi;
%p A345429 od: od:
%p A345429 ansu:=[op(ansu),tu];
%p A345429 ansv:=[op(ansv),tv];
%p A345429 ansb:=[op(ansb),tb];
%p A345429 od:
%p A345429 ansu; # the present sequence
%p A345429 ansv; # A345430
%p A345429 ansb; # A345431
%p A345429 # for A345432, A345433, A345434, omit the "igcd(x,y)=1" test
%o A345429 (Python)
%o A345429 from sympy.core.numbers import igcdex
%o A345429 def A345429(n): return sum(abs(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_, Jun 22 2021
%Y A345429 Cf. A345415-A345434.
%K A345429 nonn
%O A345429 1,3
%A A345429 _N. J. A. Sloane_, Jun 22 2021