A345431 -id:A345431 - cp's OEIS frontend

A345429 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) = sum of the values of |u|.

0, 1, 4, 7, 16, 19, 37, 49, 70, 82, 127, 145, 208, 235, 277, 325, 433, 472, 607, 667, 757, 832, 1030, 1102, 1291, 1399, 1582, 1708, 2023, 2119, 2479, 2671, 2911, 3103, 3409, 3571, 4084, 4327, 4669, 4909, 5539, 5737, 6430, 6760, 7162, 7525, 8353, 8641, 9415, 9787

Offset: 1

N. J. A. Sloane, Jun 22 2021

nonn

Minimal means minimize u^2+v^2. We follow Maple, PARI, etc., in setting u=0 and v=1 when x=y.

It would be nice to have b-files for this and related sequences (as listed in cross-references). The present sequence is especially interesting. What is its rate of growth?

Cf. A345415-A345434.

mygcd:=proc(a,b) local d,s,t; d := igcdex(a,b,`s`,`t`); [a,b,d,s,t]; end;
ansu:=[]; ansv:=[]; ansb:=[];
for N from 1 to 80 do
tu:=0; tv:=0; tb:=0;
for x from 1 to N do
for y from 1 to N do
if igcd(x,y)=1 then
   tu:=tu+abs(mygcd(x,y)[4]);
   tv:=tv+abs(mygcd(x,y)[5]);
   tb:=tb+mygcd(x,y)[4]^2 + mygcd(x,y)[5]^2;
fi;
od: od:
ansu:=[op(ansu),tu];
ansv:=[op(ansv),tv];
ansb:=[op(ansb),tb];
od:
ansu; # the present sequence
ansv; # A345430
ansb; # A345431
# for A345432, A345433, A345434, omit the "igcd(x,y)=1" test

from sympy.core.numbers import igcdex
def A345429(n): return sum(abs(u) for u, v, w in (igcdex(x,y) for x in range(1,n+1) for y in range(1,n+1)) if w == 1) # Chai Wah Wu, Jun 22 2021

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.

Original entry on oeis.org

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)

cp's OEIS Frontend

A345429 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) = sum of the values of |u|.

Views

Author

Keywords

Comments

Crossrefs

Programs

Maple

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.

Views

Author

Keywords

Comments

Crossrefs

Programs

Python

cp's OEIS Frontend

A345429 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) = sum of the values of |u|.

Views

Author

Keywords

Comments

Crossrefs

Programs

Maple

Python

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

A345429 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) = sum of the values of |u|.

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.