A206853 a(1)=1, for n>1, a(n) is the least number > a(n-1) such that the Hamming distance D(a(n-1), a(n)) = 2.

1, 2, 4, 7, 11, 13, 14, 22, 26, 28, 31, 47, 55, 59, 61, 62, 94, 110, 118, 122, 124, 127, 191, 223, 239, 247, 251, 253, 254, 382, 446, 478, 494, 502, 506, 508, 511, 767, 895, 959, 991, 1007, 1015, 1019, 1021, 1022, 1534, 1790, 1918, 1982, 2014, 2030, 2038, 2042

Offset: 1

Author

Vladimir Shevelev, Feb 13 2012

Keywords

• nonn
• base

Comments

For integers a, b, denote by a<+>b the least c>=a, such that D(a,c)=b (note that, generally speaking, a<+>b differs from b<+>a). Then a(n+1)=a(n)<+>2. Thus this sequence is a Hamming analog of odd numbers 1,3,5,...

A Hamming analog of nonnegative integers is A000225 and a Hamming analog of the triangular numbers is A000975.

All terms are odious (A000069).

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A182187 (n<+>2), A207063 (starting from 0).

Cf. A000225, A205509, A205510, A205511, A205302, A205649, A205533, A122565, A206852, A000069.

Programs

Maple

read("transforms");
Hamming := proc(a,b)
        XORnos(a,b) ;
        wt(%) ;
end proc:
Dplus := proc(a,b)
        for c from a to 1000000 do
                if Hamming(a,c)=b then
                        return c;
                end if;
        end do:
        return -1 ;
end proc:
A206853 := proc(n)
        option remember;
        if n = 1 then
                1;
        else
                Dplus(procname(n-1),2) ;
        end if;
end proc: # R. J. Mathar, Apr 05 2012

Mathematica

myHammingDistance[n_, m_] := Module[{g = Max[m, n], h = Min[m, n]}, b1 = IntegerDigits[g, 2]; b2 = IntegerDigits[h, 2, Length[b1]]; HammingDistance[b1, b2]]; t = {1}; Do[If[myHammingDistance[t[[-1]], n] == 2, AppendTo[t, n]], {n, 2, 2042}]; t (* T. D. Noe, Mar 07 2012 *)
t={x=1}; Do[i=x+1; While[Count[IntegerDigits[BitXor[x,i],2],1]!=2,i++]; AppendTo[t,x=i],{n,53}]; t (* Jayanta Basu, May 26 2013 *)

PARI

next_A206853(n)={my(b=binary(n)); until(norml2(binary(n)-b)==2, n++>=2^#b & b=concat(0,b)); n}
print1(n=1); for(i=1,99,print1(","n=next_A206853(n)))  \\ M. F. Hasler, Apr 07 2012