cp's OEIS Frontend

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.

%I A345725 #7 Jun 25 2021 01:59:16
%S A345725 0,0,10,28,166,224,964,1624,3626,4934,12308,15928,33670,42828,62656,
%T A345725 85016,154016,181254,301688,364896,480428,580134,901698,1021274,
%U A345725 1412852,1655336,2149650,2503910,3518644,3847556,5247764,6093004,7339188,8291404,10135408,11018524
%N 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.
%C A345725 The factor m^2 is to ensure that a(n) is an integer.
%C A345725 A345425(n) = m*mu where mu is the mean of the values of u+v.
%C A345725 The population standard deviation sqrt(s) appears to grow linearly with n.
%o A345725 (Python)
%o A345725 from statistics import pvariance
%o A345725 from sympy.core.numbers import igcdex
%o A345725 def A345725(n):
%o A345725     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]
%o A345725     return pvariance(len(zlist)*(u+v) for u, v, w in zlist)
%Y A345725 Cf. A345425, A345687, A345688, A345689, A345690, A345691, A345692, A345693, A345694, A345695.
%K A345725 nonn
%O A345725 1,3
%A A345725 _Chai Wah Wu_, Jun 24 2021