sums of factors in GF(2)

[Antti Karttunen]

Marc LeBrun wrote:

> 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.
>
Don't worry, I will do the GF(2)[X] -thing, in my due time. (This is 
easy to do
by using the existing the 
http://www.research.att.com/~njas/sequences/a091247.scm.txt
Analogues for http://www.research.att.com/~njas/sequences/A000203 and 
similar sequences
as well...)


> 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?
>
If we could just find from 
http://www.research.att.com/~njas/sequences/A014580
two successive terms, with first A014580(k) = 1 mod 4, and the second 
A014580(k+1) = A014580(k)+2,
then by xoring them we could get 2, and the product A014580(k) x 
A014580(k+1) x 2
(where x would be GF(2)[X] multiplication: 
http://www.research.att.com/~njas/sequences/A048720 )
would satisfy the condition.
However, skimming cursorily over the first 100 terms of  
http://www.research.att.com/~njas/sequences/A058943
I didn't find such a pair. (Maybe there's a reason for that? GF(2)[X] 
secrets elide me for a monent...)

Yours,

Antti