[seqfan] A conjecture on Hamming distance

[Vladimir Shevelev]

Dear SeqFans,


A conjecture which I formulate below arose after publication sequences A209544 and A209554. In the first one, R. Mathar asks: are these related to A141174,A045390 or A007519?; in the second one,  he again asks: how relate these to A133870?  Using the definition of operation <+> (which was introduced by me in comment in A206853), I now formulate a general conjecture.
Let n>=3 be odd and k>=2. We say that n possesses a property S_k, if for every integer m from interval [0,n) with the Hamming distance D(m,n) in [2,k], there exists an integer h from (m,n) with D(m,h)=D(m,n).
Conjecture. Odd n>3 possesses the property S_k iff n has the form n=2^(2*k-1)*t+1.
Example. Let k=2, t=1. Then n=9=(1001)_2. All numbers m from [0,9) with D(m,9)=2 are 0,3,5.
For m=0, we can take h=3, since 3 from (0,9) and D(0,3)=2; for m=3, we can take h=5, since 5 from (3,9) and D(3,5)=2; for m=5,  we can take h=6, since 6  from (5,9) and D(5,6)=2.

You are welcome to prove (disprove) this conjecture!

Regards,
Vladimir


 Shevelev Vladimir‎