Congruent Products Under XOR; Fibbinary Numbers

[Paul D. Hanna]

Seqfans, 
      For A003714 (Fibbinary numbers), Reinhard Zumkeller gives the
definition:  
"Numbers m such that m XOR 2*m = 3*m."
  
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? 
 
So, the definition of this family of sequences may be clarified to be:  
"Numbers n such that: p*n XOR q*n = (p XOR q)*n." 
  
Q: Are these sequences all distinct?
A: No, there are sequences that are equal for different values of p and
q. 
 
Here I make two simple observations. 
(1) These sequences are equal: 
{n such that p*n XOR q*n = (p XOR q)*n} = 
{n such that 2^n*p*n XOR 2^n*q*n = 2^n*(p XOR q)*n}. 
(2) These sequences are also equal:  
{n such that p*n XOR q*n = (p XOR q)*n} = 
{n such that p*n XOR (p XOR q)*n = q*n} = 
{n such that (p XOR q)*n XOR q*n = p*n}. 
  
But there are other (more complex) congruences besides these for distinct
p and q.  
Let us for brevity denote these sequences by {p, q, r}, where r = p XOR
q.
 
Q: How can one determine if sequences {p1,q1,r1} and {p2,q2,r2} are
equivalent? 
A: Unkown ... 
 
Below I list 5 groups of sequences generated by different {p, q, r} that
appear 
to be equivalent within each group; only those with odd GCD(p,q) are
listed.  
Each sequence was generated using n=1..65536, and found to be equal to
each other in the same 

group, so it is likely that the equivalence will continue to hold for
n>2^16. 
 
Can anyone figure out when {p1,q1,r1} and {p2,q2,r2} are equivalent? 
Is there any references to this in the literature? 
 
Thanks,
       Paul
------------------------------------------------------------  
{1,6,7}={3,4,7}={3,5,6}=...
Numbers n such that: n XOR 6*n = 7*n.
= A048715 ; Binary expansion matches ((0)*001)*(0*); or, 
Zeckendorf-like expansion of n using recurrence f(n) = f(n-1) + f(n-3). 
------------------------------------------------------------  
{1,14,15}={3,9,10}={3,13,14}={5,9,12}={5,11,14}={6,11,13}=
{7,9,14}={7,10,13}={7,11,12}={12,21,25}={12,37,41}=...
Numbers n such that: n XOR 14*n = 15*n.
= A048718 ; Binary expansion matches ((0)*0001)*(0*); or, 
Zeckendorf-like expansion of n using recurrence f(n) = f(n-1) + f(n-4).
------------------------------------------------------------  
{1,20,21}={5,16,21}={5,17,20}=...
Numbers n such that: n XOR 20*n = 21*n.
= NEW SEQUENCE A115422:
1,2,3,4,6,8,9,12,16,18,24,32,33,36,48,64,65,66,67,72,73,96,97,128,
129,130,131,132,134,144,146,192,193,194,195,256,257,258,259,
260,262,264,265,268,288,289,292,384,385,386,387,388,390,
------------------------------------------------------------ 
{1,30,31}={3,21,22}={3,25,26}={3,29,30}={5,19,22}={5,25,28}=
{5,27,30}={6,19,21}={7,17,22}={7,18,21}={7,19,20}={7,25,30}=
{7,26,29}={7,27,28}={9,19,26}={9,21,28}={9,23,30}={10,19,25}=
{11,17,26}={11,18,25}={11,19,24}={11,21,30}={11,22,29}={11,23,28}=
{12,23,27}={13,17,28}={13,19,30}={13,20,25}={13,21,24}={13,23,26}=...
Numbers n such that: n XOR 30*n = 31*n.
= NEW SEQUENCE A115423:
1,2,4,8,16,32,33,64,65,66,128,129,130,132,256,257,258,260,264,512,
513,514,516,520,528,1024,1025,1026,1028,1032,1040,1056,1057,2048,
2049,2050,2052,2056,2064,2080,2081,2112,2113,2114,4096,
------------------------------------------------------------  
{1,62,63}={3,41,42}={3,45,46}={3,53,54}={3,57,58}={3,61,62}=
{5,35,38}={5,57,60}={5,59,62}={6,35,37}={6,43,45}={6,51,53}=
{7,33,38}={7,35,36}={7,41,46}={7,42,45}={7,43,44}={7,49,54}=
{7,50,53}={7,51,52}={7,57,62}={7,58,61}={7,59,60}={9,49,56}=
{9,51,58}={9,53,60}={9,55,62}={10,35,41}={10,39,45}={11,33,42}=
{11,34,41}={11,35,40}={11,37,46}={11,39,44}={11,50,57}={11,51,56}=
{11,53,62}={11,54,61}={11,55,60}={12,39,43}={13,35,46}={13,38,43}=
{13,39,42}={13,51,62}={13,52,57}={13,53,56}={13,54,59}={13,55,58}=...
Numbers n such that: n XOR 62*n = 63*n.
= NEW SEQUENCE A115424:
1,2,4,8,16,32,64,65,128,129,130,256,257,258,260,512,513,514,516,520,
1024,1025,1026,1028,1032,1040,2048,2049,2050,2052,2056,2064,2080,
4096,4097,4098,4100,4104,4112,4128,4160,4161,8192,8193,8194,8196,
------------------------------------------------------------  
END