A163188 - OEIS

%I #5 Aug 24 2013 12:12:50

%S 4,18,48,100,150,180,294,448,490,588,648,900,960,1134,1210,1584,1620,

%T 2028,2100,2178,2548,2904,3150,3388,3630,3718,3840,4624,5040,5070,

%U 5508,5850,6084,6468,6498,6760,7098,7600,8670,8820,9900,9984,10164,11638

%N Numbers of the form n = r*s = (r+s)*t with gcd(r+s,t) = 1.

%C Also numbers of the form n = u*v*(u+v)^2 with gcd(u,v) = 1. The connection to the definition is given by r = u*(u+v), s = v*(u+v), t = u*v, resp. u = gcd(r,t), v = gcd(s,t).

%C Also "primitive" members of A139719: With k as in the definition of A139719, we additionally require that gcd(k+n/k, n/(k+n/k)) = 1.

%C 100 has a non-primitive solution with k=10, resp. (r,s,t) = (10,10,5), resp. (u,v) = (5,5). It is included because there is also the primitive solution k=5, resp. (r,s,t) = (5,20,4), resp. (u,v) = (1,4).

%C 8820 has two primitive solutions: k=21, resp. (r,s,t) = (21,420,20), resp. (u,v) = (1,20) and k=70, resp. (r,s,t) = (70,126,45), resp. (u,v) = (5,9).

%e 4 is in the sequence because 4 = 2*2 = (2+2)*1, gcd(2+2,1)=1.

%e 18 is in the sequence because 18 = 3*6 = (3+6)*2, gcd(3+6,2)=1.

%e 48 is in the sequence because 48 = 4*12 = (4+12)*3, gcd(4+12,3)=1.

%e 16 = 4*4 = (4+4)*2 is not sufficient to make 16 a member of the sequence because gcd(4+4,2)=2.

%o (PARI) L=10000;v=[];for(r=1,L^(1/3),for(s=1,r,if(gcd(r,s)==1, n=r*s*(r+s)^2; if(n>L,break);if(n==8820,print([r,s]));v=concat(v,n))));vecsort(eval(Set(v)))

%Y Cf. A139719.

%K nonn

%O 1,1

%A _Hagen von Eitzen_, Jul 22 2009