Another Sequence Related To Binary Palindromes

[Leroy Quet]

I also just submitted this:

%S A000001 0,3,0,1,0,1,0,1,0,0,0,0,0,1,0,1,0,0,0,1
%N A000001 a(n) = the largest integer such that n^k is palindromic in binary for all nonnegative integers k that are <= a(n).
%C A000001 a(2n) = 0 for all n.
%e A000001 The powers of 3 are, when written in binary: 1, 11, 1001, 11011, 1010001,... Now, 3^k written in binary is palindromic for k = 0,1,2, and 3, but not for k=4. So a(3) = 3.
%O A000001 2
%K A000001 ,more,nonn,

What are the n's where a(n) is a record?

Thanks,
Leroy Quet