Neil B. points out that Phil's PARI code misses eleven entries in http://oeis.org/A129935/b129935.txt, and suggests 13*cand (or larger) instead of 2*cand in

default(realprecision, 500); c=contfrac(log(2)/2); for(n=2,#c, cand=contfracpnqn(vecextract(c, 2^n-1))[1, 1];forstep(m=cand, 2*cand, cand, if(ceil(2/(2^(1/m)-1)) != floor(2*m/log(2)),print(m))))

13, and even 20, is in turn insufficient for longer tables. 20 gives 638215591788186030688409<<5153>>7190350518178849343531561 as A(1520), but 26 gives it as A(1527).  Is there any finite alternative to 26 that works indefinitely?  If not, what is a correct algorithm here?  This may relate to the startling interludes of slow, arithmetic growth at, e.g., A(56)-A(68).

Would it depend on the next term in the CF? Could the numerator and denominator ever exceed the values attained by the next convergent? Of not, then that should provide an upper limit. I can't say I was ever happy with that 2, it's just that I knew I was missing some even numbers in the listed terms, and the 2 would catch them. Phil -- () ASCII ribbon campaign () Hopeless ribbon campaign /\ against HTML mail /\ against gratuitous bloodshed [stolen with permission from Daniel B. Cristofani]