On 2013-02-27, at 3:05 PM, Bill Gosper wrote: Sum[2^k/(1 + z^2^k), {k, -Infinity, Infinity}] == 1/Log[z] Can somebody tell me where? --rwg And, as Neil empiricizes, Sum[2^k/(1 + z^2^k), {k, 0, Infinity}] == 1/(z-1) MRob>This reminds me of something Neil Sloane mentions in his OEIS talks, namely that the function f(n) = floor(2n/log(2)) has the same value as the function g(n) = ceiling(2/(2^(1/n)-1)) (Sloane's sequence A78608) for all integer n from 1 to [7]77451915729367, but differs at n=77451915729368 (and at some other larger values, seeoeis.org/A129935). -------------------------- 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). Here is equivalent Mma code: $MaxExtraPrecision = 999999999; Select[Flatten[ Transpose[Outer[Times, Range[26], Denominator@Convergents[2/Log[2], 9999]]]], Floor[2*#/Log[2]] != Ceiling[2/(2^(1/#) - 1)] &]; This should really say $MaxExtraPrecision = ∞ but that tickles a numerics bug in the current version (9.0.1). --rwg