A345695 -id:A345695 - cp's OEIS frontend

A345696 For 1<=x<=n, 1<=y<=n with gcd(x,y)=1, write 1 = gcd(x,y) = ux+vy 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

Chai Wah Wu, Jun 24 2021

nonn

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.

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

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)

A345725 For 1<=x<=n, 1<=y<=n with gcd(x,y)=1, write 1 = gcd(x,y) = ux+vy 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.

Original entry on oeis.org

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

Chai Wah Wu, Jun 24 2021

nonn

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.

Cf. A345425, A345687, A345688, A345689, A345690, A345691, A345692, A345693, A345694, A345695.

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)

cp's OEIS Frontend

A345696 For 1<=x<=n, 1<=y<=n with gcd(x,y)=1, write 1 = gcd(x,y) = ux+vy 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.

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A345725 For 1<=x<=n, 1<=y<=n with gcd(x,y)=1, write 1 = gcd(x,y) = ux+vy 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.

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Author

Keywords

Comments

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Programs

Python

cp's OEIS Frontend

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.

Views

Author

Keywords

Comments

Crossrefs

Programs

Python

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.

Views

Author

Keywords

Comments

Crossrefs

Programs

Python

A345696 For 1<=x<=n, 1<=y<=n with gcd(x,y)=1, write 1 = gcd(x,y) = ux+vy 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.

A345725 For 1<=x<=n, 1<=y<=n with gcd(x,y)=1, write 1 = gcd(x,y) = ux+vy 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.