First if you check using f(x,y) > (x+1)/y + (y+1)/x
you find that
f(y,y(y+1)/x) comes out to the same thing (algebra not shown.)
so if you find one solution you get infinitely many.
I found two families based on
f(2,3)=3 and f(1,2)=4
but no others up to 10.
The first few terms that yield 3 are
(2,2)
(2,3)
(3,6)
(6,14)
(14,35)
(35,90)
The first few terms that yield 4 are
(1,1)
(1,2)
(2,6)
(6,21)
(21,77)
(77,286)
In trying to find other yields I checked up to 10 using 100 as the smaller number. There are sum interesting near misses such as
f(88/870) = 4399/440 (close to 10) which may warrant further investigation.
Edited because I forgot the first pair of each list.
Edited on August 30, 2015, 8:51 pm

Posted by Jer
on 20150830 16:47:23 