Integer Division Loops

Haskell, 227 203 bytes

17 bytes saved thanks to Οurous

(%)=div
u x|odd x=x%2+1|1>0=x%2
[0,b]!l=[b]!l
[b,0]!l=[b]!l
t!l|elem t l=[dropWhile(/=t)l]
t@[a,b]!l|q<-l++[t]=[a%2,u a+b]!q++[u b+a,b%2]!q
f x|q<-[x%2,u x]![[x]]=[a|(a,b)<-zip q[0..],notElem a$take b q]

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Clean, 267 ... 167 bytes

import StdEnv
f s l|any((==)s)l=[dropWhile((<>)s)l]#[h,t:_]=s
|1>h*t=f[max h t]l=f[h/2,(h+1)/2+t](l++[s])++(f[(t+1)/2+h,t/2](l++[s]))
@n=removeDup(f[n/2,(n+1)/2][[n]])

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Ungolfed:

loops num
    = removeDup (half (divide num) [[num]])
where
    divide num
        = [num/2, (num+1)/2]
    half [_, 0] list
        = list
    half [0, _] list
        = list
    half [a, b] list
        | isMember [a, b] list
            = [dropWhile ((<>) [a, b]) list]
        # [u, v: _]
            = divide a
        # [x, y: _]
            = divide b
        = (half [u, v+b] (list ++ [a, b])) ++ (half [a+y, x] (list ++ [a, b]))