A345430 -id:A345430

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A345695

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 |v| and m is the number of such values.

+10
3

0, 2, 10, 24, 110, 152, 656, 1198, 2714, 3632, 9512, 13082, 27274, 34474, 51416, 71168, 128704, 152430, 253648, 311636, 412538, 495234, 766258, 877438, 1217102, 1420616, 1843136, 2170622, 3039784, 3342200, 4551830, 5284110, 6360830, 7182594, 8780236, 9608714

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

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

A345430(n) = m*mu where mu is the mean of the values of |v|.

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

LINKS

Table of n, a(n) for n=1..36.

PROG

(Python)

from statistics import pvariance

from sympy.core.numbers import igcdex

def A345695(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)*abs(v) for u, v, w in zlist)

CROSSREFS

Cf. A345430, A345687, A345688, A345689, A345690, A345691, A345692, A345693, A345694.

KEYWORD

nonn

AUTHOR

Chai Wah Wu, Jun 24 2021

STATUS

approved

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|.

+10
2

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,3

COMMENTS

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?

LINKS

Table of n, a(n) for n=1..50.

MAPLE

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