cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-2 of 2 results.

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.

Original entry on oeis.org

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
Offset: 1

Views

Author

Chai Wah Wu, Jun 24 2021

Keywords

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.

Crossrefs

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

Programs

  • 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)

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|.

Original entry on oeis.org

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

Views

Author

N. J. A. Sloane, Jun 22 2021

Keywords

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?

Crossrefs

Cf. A345415-A345434.

Programs

  • 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
  • Python
    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
Showing 1-2 of 2 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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