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.

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.

Links

Table of n, a(n) for n=1..30.

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)

Crossrefs

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

Sequence in context: A185985 A358986 A239477 * A066527 A343385 A103423

Adjacent sequences: A345693 A345694 A345695 * A345697 A345698 A345699

Keyword

nonn

Author

Chai Wah Wu, Jun 24 2021

Status

approved