sums of factors in GF(2)

[Marc LeBrun]

Recently I've been trying to exhort the Sequence Phanatiques to 
compute analogs of the sum-of-prime-factors (with and without 
multiplicity) in other arithmetics, such as Gaussian integers, GF(2), etc.

I was wondering, specifically about GF(2), summing (ie XORing) the 
prime factors of N with multiplicity:

Noting that only the square-free part of N matters, since the square 
parts sum to 0...

A. Aside from the perfect squares (eg 5) are there any other N that 
sum to 0?  Can they be characterized?

B. If some sum S occurs for any N then it occurs infinitely, for all 
the square multiples of N.  Does every value of S occur?