The problem is not possible.
Suppose there is a term n such that S(n)=0
That means 0 = S(n2)  3/S(n1)
so S(n1)*S(n2)=3
But S(n1) = S(n3)  3/S(n2)
and substituting in gives
[S(n3)  3/S(n2)]*S(n2)=3
S(n3)*S(n2)=0 < Algebra error. This should be 6.
Now S(n2) cannot be 0, so S(n3)=0
This means S(n1)=03/S(n2)
meaning S(n2)*S(n1)=3
Which is a contradiction.
So there is no n such that S(n)=0.
***** Edit *****
Daniel's solution seems valid and a quick check with Excel seems to confirm this. We posted at nearly the same time so I didn't see his until I hit enter. I think something must be wrong with the above. I wonder what.
Now that I've got S(n3)*S(n2)=6 I could probably show S(n(a+1))*S(na)=3a and work backwards to give the same solution as Daniel.
Edited on January 28, 2016, 10:46 am

Posted by Jer
on 20160128 09:34:50 