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A273039 Numbers k such that the following process converges to zero: x(0)=k, x(i+1) = x(i) XOR ror(x(i)) XOR rol(x(i)); see the Comments section for details.

%I A273039 #20 Jun 11 2025 12:33:00
%S A273039 0,5,6,9,24,29,34,40,43,45,48,51,54,57,65,66,68,71,75,77,80,83,86,89,
%T A273039 90,92,101,102,111,129,130,135,139,141,153,154,159,180,189,198,204,
%U A273039 209,216,219,226,231,232,238,257,260,263,267,272,275,277,278,282,284,297
%N A273039 Numbers k such that the following process converges to zero: x(0)=k, x(i+1) = x(i) XOR ror(x(i)) XOR rol(x(i)); see the Comments section for details.
%C A273039 Numbers k such that the following process converges to zero: x(0)=k, x(i+1) = x(i) XOR ror(x(i)) XOR rol(x(i)), where XOR is the binary exclusive-or operator, ror(x)=A038572(x) is x rotated one binary place to the right, and similarly rol(x)=A006257(k) is x rotated one binary place to the left.
%e A273039 n=5:  x(0)=5, x(1) = 5 xor 6 xor 3 = 0.
%e A273039 n=6:  x(0)=6, x(1) = 6 xor 5 xor 3 = 0.
%e A273039 n=9:  x(0)=9, x(1) = 9 xor 12 xor 3 = 6, x(2)=0.
%e A273039 n=10: x(0)=10, x(1) = 10 xor 5 xor 5 = 10, and x(i)=10 for i>1.
%e A273039 n=17: x(0)=17, x(1) = 17 xor 24 xor 3 = 10, and x(i)=10 for i>1.
%e A273039 So 5, 6, 9 are in the sequence, 10 and 17 are not.
%t A273039 Select[Range[0, 300], Nest[BitXor[BitXor[#, FromDigits[ RotateRight[ IntegerDigits[#, 2]], 2]], FromDigits[ RotateLeft[ IntegerDigits[#, 2]], 2]] &, #, 120] == 0 &] (* _Michael De Vlieger_, May 14 2016 *)
%o A273039 (Python)
%o A273039 def ROR(n):                # returns A038572(n)
%o A273039     BL = len(bin(n))-2
%o A273039     return (n>>1) + ((n&1) << (BL-1))
%o A273039 def ROL(n):                # returns A006257(n)
%o A273039     BL = len(bin(n))-2
%o A273039     return (n*2) - (1<<BL) + 1
%o A273039 for n in range(1000):
%o A273039     X = n
%o A273039     Xs = []
%o A273039     while not (X in Xs):
%o A273039         Xs.append(X)
%o A273039         if X==0:
%o A273039             print(n, end=', ')
%o A273039             break
%o A273039         X = X ^ ROR(X) ^ ROL(X)
%Y A273039 Cf. A038572, A006257.
%K A273039 nonn,base
%O A273039 1,2
%A A273039 _Alex Ratushnyak_, May 13 2016