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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^2+v^2.
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%I #10 Jun 25 2021 00:31:56

%S 1,3,9,15,41,47,119,171,281,333,623,755,1233,1419,1799,2231,3319,3705,

%T 5215,5943,7075,7953,10665,11665,14467,15983,18949,21081,26479,28239,

%U 34803,38403,43203,46779,53215,56555,67885,73115,81015,86711,101891,106999,124753,133461,144273

%N 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^2+v^2.

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

%o (Python)

%o from sympy.core.numbers import igcdex

%o def A345431(n): return sum(u**2+v**2 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 Cf. A345415-A345434.

%K nonn

%O 1,2

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