login
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.
1
0, 0, 10, 28, 846, 1080, 13524, 28336, 101274, 130086, 526116, 796704, 2121646, 2676676, 5103216, 7545320, 16863936, 20080798, 39983568, 51986376, 78689204, 96323998, 175534714, 207346098, 324942572, 386288432, 560665370, 693425934, 1087095852, 1220707044
OFFSET
1,3
COMMENTS
The factor m^2 is to ensure that a(n) is an integer.
A345431(n) = m*mu where mu is the mean of the values of u^2+v^2.
s^(1/4) appears to grow linearly with n.
PROG
(Python)
from statistics import pvariance
from sympy.core.numbers import igcdex
def A345696(n):
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]
return pvariance(len(zlist)*(u**2+v**2) for u, v, w in zlist)
KEYWORD
nonn
AUTHOR
Chai Wah Wu, Jun 24 2021
STATUS
approved