(suggested new terminology)
For all powers of 2, w, call the smallest (2^n)*3 > w, for n >= 0, the sidekick of w, and denote it by s(w). Then the following 4 facts are easily proved:
(1) for each n >= 1, s(2^n) = (3/2)*(2^n) (and therefore (because of the 3/2 factor), as noted by Melvin Peralta and Miriam Ong Ante for OEIS sequence A007283, the sidekicks constitute the average of consecutive powers of 2, starting with 2^1);
(2) for n > 1, s(2^n) is the one and only number strictly between 2^n and 2^(n+1) whose
bigomega value is equal to the bigomega value of (2^n);
(3) for n > 1, like both 2^n and 2^(n+1), s(2^n) is the smallest number having its prime signature;
(4) for n > 1, phi(s(2^n)) = phi(2^n).
[Number Theory]