OFFSET
1,1
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).
EXAMPLE
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.
PROG
(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)))
CROSSREFS
KEYWORD
nonn
AUTHOR
Hagen von Eitzen, Jul 22 2009
STATUS
approved