Congruent Products Under XOR; Fibbinary Numbers

[Paul D. Hanna]

Dean (and Seqfans), 
      Thanks for catching my mistake.  
It seems obvious (now) that the conjecture is false - in fact, 
there are so many exceptions I wonder how I missed them. 
A better conjecture may be: 
there exists an infinite set of integers n that satisfy 
p*n XOR q*n = (p XOR q)*n for all positive integers p,q, r. 
I find no exceptions to this guess. 
 
Yet my focus was primarily on what different values of {p,q,r} 
define the same sequence that satisfies p*n XOR q*n = r*n 
when GCD(p,q) is odd.  This remains a mystery to me. 
 
Perhaps Ralf Stephan's comment offers a clue; 
the binary mask, such as ((0)*0001)*(0*) that match A048718(n),  
may determine which {p, q, r} define equivalent sequences. 
 
Thanks, 
      Paul 
 
> > As a generalization, consider the sequences defined by:
> > "Numbers n such that: p*n XOR q*n = r*n."  for positive integers 
> p, q, r.
> >
> > Q: For what p, q, r, does the above definition generate sequences 
> not {0}?
> > A: Iff  r = p XOR q (conjecture).
> >
> > Can anyone prove this conjecture?
> 
> It's not true.  Let  p=2,  q=3,  and  r=5.  Then  n=3  is in the 
> sequence,
> since  2*3 XOR 3*3 = 6 XOR 9 = 15 = 5*3.  But  2 XOR 3 = 1,  not 5.
> 
> Dean Hickerson