%I #14 May 26 2017 21:24:58
%S 1,1,2,3,3,3,4,5,5,5,6,6,7,7,7,8,8,9,9,9,9,10,10,11,11,11,11,12,12,12,
%T 13,13,13,13,14,14,14,15,15,15,15,16,16,16,17,17,17,17,17,18
%N Order of largest subset of the positive rationals with neither the sum of numerators nor of denominators exceeding n.
%e For n = 1, the largest subset is { 1/1 }, so a(1) = 1; for n = 2, the same; for n = 3, largest subsets are { 1/1, 2/1 } and { 1/2, 1/1 }, so a(3) = 2; for n = 4, the largest subset is { 1/2, 1/1, 2/1 }, so a(4) = 3; ...; for n = 16, the largest subset is { 1/4, 1/3, 1/2, 2/3, 1/1, 2/1, 3/1, 4/1 } (or swap 2/3 for 3/2), so a(16)=8, etc.
%o (Haskell)
%o f = go 0 2
%o where
%o go a r n
%o  n >= c = go (a+t) (r+1) (nc)
%o  n >= r*div n r + m = a + 2*div n r + 1
%o  n >= r*div n r + m' + 1 = a + 2*div n r + 1
%o  otherwise = a + 2*div n r
%o where
%o t = totient r
%o c = div (r*t) 2
%o m = midnum r
%o m' = midnum (r1)
%o midnum r = head [aa<[div (r+1) 2..], gcd a r==1]
%K nonn
%O 1,3
%A _Carl Edman_, May 23 2017
