[math-fun] sums of factors in GF(2)

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?