A345696 - OEIS

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 #7 Jun 25 2021 01:59:29

%S 0,0,10,28,846,1080,13524,28336,101274,130086,526116,796704,2121646,

%T 2676676,5103216,7545320,16863936,20080798,39983568,51986376,78689204,

%U 96323998,175534714,207346098,324942572,386288432,560665370,693425934,1087095852,1220707044

%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) = 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 The factor m^2 is to ensure that a(n) is an integer.

%C A345431(n) = m*mu where mu is the mean of the values of u^2+v^2.

%C s^(1/4) appears to grow linearly with n.

%o (Python)

%o from statistics import pvariance

%o from sympy.core.numbers import igcdex

%o def A345696(n):

%o 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 return pvariance(len(zlist)*(u**2+v**2) for u, v, w in zlist)

%Y Cf. A345431, A345687, A345688, A345689, A345690, A345691, A345692, A345693, A345694, A345695.

%K nonn

%O 1,3

%A _Chai Wah Wu_, Jun 24 2021