A345724 For 1<=x<=n, 1<=y<=n, write gcd(x,y) = u*x+v*y with u,v minimal; a(n) = n^4*s, where s is the population variance of the values of u+v.

0, 0, 14, 48, 250, 452, 1578, 2816, 6120, 9556, 20220, 28476, 54596, 75092, 111050, 155120, 253852, 323792, 497054, 624700, 828476, 1049584, 1510824, 1792476, 2397166, 2924432, 3736358, 4469884, 5919800, 6804500, 8811122, 10401536, 12541844, 14621072, 17574850

Offset: 1

Author

Chai Wah Wu, Jun 24 2021

Keywords

• nonn

Comments

The factor n^4 is to ensure that a(n) is an integer.

A345428(n) = n^2*mu where mu is the mean of the values of u+v.

The population standard deviation sqrt(s) appears to grow linearly with n.

Crossrefs

Cf. A345428, A345687, A345688, A345689, A345690, A345691.

Programs

Python

from statistics import pvariance
from sympy.core.numbers import igcdex
def A345724(n): return pvariance(n**2*(u+v) for u, v, w in (igcdex(x,y) for x in range(1,n+1) for y in range(1,n+1)))