A345725 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+v and m is the number of such values.

0, 0, 10, 28, 166, 224, 964, 1624, 3626, 4934, 12308, 15928, 33670, 42828, 62656, 85016, 154016, 181254, 301688, 364896, 480428, 580134, 901698, 1021274, 1412852, 1655336, 2149650, 2503910, 3518644, 3847556, 5247764, 6093004, 7339188, 8291404, 10135408, 11018524

Offset: 1

Author

Chai Wah Wu, Jun 24 2021

Keywords

• nonn

Comments

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

A345425(n) = m*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. A345425, A345687, A345688, A345689, A345690, A345691, A345692, A345693, A345694, A345695.

Programs

Python

from statistics import pvariance
from sympy.core.numbers import igcdex
def A345725(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+v) for u, v, w in zlist)