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A345696 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) = m^2*s, where s is the population variance of the values of u^2+v^2 and m is the number of such values.

%I A345696 #7 Jun 25 2021 01:59:29
%S A345696 0,0,10,28,846,1080,13524,28336,101274,130086,526116,796704,2121646,
%T A345696 2676676,5103216,7545320,16863936,20080798,39983568,51986376,78689204,
%U A345696 96323998,175534714,207346098,324942572,386288432,560665370,693425934,1087095852,1220707044
%N A345696 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) = m^2*s, where s is the population variance of the values of u^2+v^2 and m is the number of such values.
%C A345696 The factor m^2 is to ensure that a(n) is an integer.
%C A345696 A345431(n) = m*mu where mu is the mean of the values of u^2+v^2.
%C A345696 s^(1/4) appears to grow linearly with n.
%o A345696 (Python)
%o A345696 from statistics import pvariance
%o A345696 from sympy.core.numbers import igcdex
%o A345696 def A345696(n):
%o A345696     zlist = [z for z in (igcdex(x,y) for x in range(1,n+1) for y in range(1,n+1)) if z[2] == 1]
%o A345696     return pvariance(len(zlist)*(u**2+v**2) for u, v, w in zlist)
%Y A345696 Cf. A345431, A345687, A345688, A345689, A345690, A345691, A345692, A345693, A345694, A345695.
%K A345696 nonn
%O A345696 1,3
%A A345696 _Chai Wah Wu_, Jun 24 2021