A163188 Numbers of the form n = r*s = (r+s)*t with gcd(r+s,t) = 1.

4, 18, 48, 100, 150, 180, 294, 448, 490, 588, 648, 900, 960, 1134, 1210, 1584, 1620, 2028, 2100, 2178, 2548, 2904, 3150, 3388, 3630, 3718, 3840, 4624, 5040, 5070, 5508, 5850, 6084, 6468, 6498, 6760, 7098, 7600, 8670, 8820, 9900, 9984, 10164, 11638

Offset: 1

Author

Hagen von Eitzen, Jul 22 2009

Keywords

• nonn

Comments

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).

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.

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).

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).

Examples

			4 is in the sequence because 4 = 2*2 = (2+2)*1, gcd(2+2,1)=1.
18 is in the sequence because 18 = 3*6 = (3+6)*2, gcd(3+6,2)=1.
48 is in the sequence because 48 = 4*12 = (4+12)*3, gcd(4+12,3)=1.
16 = 4*4 = (4+4)*2 is not sufficient to make 16 a member of the sequence because gcd(4+4,2)=2.
		

Crossrefs

Cf. A139719.

Programs

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)))