log(m)'s close to integers, various log bases

[Neil Fernandez]

In message <3E2C2ECD.7030104 at kspaint.com>, Robert G. Wilson v
<rgwv at kspaint.com> writes
>Et al,
>
>       The sequence begins:
>2, 3, 7, 20, 148, 403, 1096, 1097, 2980, 2981, 8103, 59874, 162755, 442413, 
>1202604, 8886110, 8886111,
>I used the Mathematica coding of:
>a = 1; Do[d = Abs[ N[ Round[ Log[n]] - Log[n], 24]]; If[d < a, Print[n]; a = d], 
>{n, 2, 10^6}]. I am sure that there must be a better way.
>16-Log(8886111)=0.0000000539...

3269017

14-Log(1202604)= 0.0000002363
15-Log(3269017)= 0.0000001139

I tried this with some other bases.

Phi gives
2, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322, 521, 843, 1364, 2207,
3571, 5778, 9349, 15127, 24476, 39603, 64079, 103682, 167761, 271443,
439204, 710647

Pi gives 2, 3, 10, 31, 306, 9488, 9499, 29808, 29809, 93648

For e, the sign of the difference between each log and the nearest
integer runs:
1, -1, -1, -1, -1, -1, 1, -1, 1, -1, -1, 1, -1, -1, -1, 1,

For pi: -1, 1, -1, -1, -1, 1, -1, -1, -1

For phi: 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1,
1, -1, 1, -1, 1, -1, 1, -1, 1

Does the phi sequence ever stop alternating between 1 and -1?

If it doesn't, is there any other real number for which the sequence
alternates in this way? (I conjecture that it doesn't. But if its
sequence does stop alternating, there still might be a number for which
the sequence doesn't. I conjecture, however, that there isn't).

Is there any real number which, used as a base for generating the first
type of sequence, yields a sequence where all terms belong to pairs,
i.e. a(2k) = a(2k-1)+1 for all positive integers k?

Is every sequence of 1s and -1s yielded by *some* number?

Is any sequence of 1s and -1s yielded by more than one number?

Neil

-- 
Neil Fernandez