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A few preliminary pages (most of these need to be updated!)
Sequence of the Day for January 4
A198683: Number of distinct values taken by
(with
’s and
pairs of parentheses inserted in all possible nontrivial ways) where
is the
imaginary unit and
denotes the
principal value of the
power function.
-
{ 1, 1, 2, 3, 7, 15, 34, 77, ... }
In the case of say,
, we’re considering the expressions
-
| i ^ (i ^ (i ^ i)), i ^ ((i ^ i) ^ i) = ((i ^ i) ^ i) ^ i, (i ^ i) ^ (i ^ i) = (i ^ (i ^ i)) ^ i, |
viz.
-
| i (i (i i ) ), i ((i i ) i ) = ((i i ) i ) i, (i i ) (i i ) = (i (i i ) ) i, |
and whether or not these expressions result in distinct values or not. As it turns out, the second and third expressions give the same value
(approximately
4.81047738...), while the fourth and fifth expressions both result in
| (exp ( − π / 2)) ^ (exp ( − π / 2)) |
, (approximately
0.721418...). (By the way, only the first of these expressions results in a number with a nonzero imaginary part.)
This is the 