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

0, 5, 6, 9, 24, 29, 34, 40, 43, 45, 48, 51, 54, 57, 65, 66, 68, 71, 75, 77, 80, 83, 86, 89, 90, 92, 101, 102, 111, 129, 130, 135, 139, 141, 153, 154, 159, 180, 189, 198, 204, 209, 216, 219, 226, 231, 232, 238, 257, 260, 263, 267, 272, 275, 277, 278, 282, 284, 297

Offset

1,2

Comments

Numbers n such that the following process converges to zero: x(0)=n, 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(n) is x rotated one binary place to the left.

Links

Table of n, a(n) for n=1..59.

Example

n=5:  x(0)=5, x(1) = 5 xor 6 xor 3 = 0.
n=6:  x(0)=6, x(1) = 6 xor 5 xor 3 = 0.
n=9:  x(0)=9, x(1) = 9 xor 12 xor 3 = 6, x(2)=0.
n=10: x(0)=10, x(1) = 10 xor 5 xor 5 = 10, and x(i)=10 for i>1.
n=17: x(0)=17, x(1) = 17 xor 24 xor 3 = 10, and x(i)=10 for i>1.
So 5, 6, 9 are in the sequence, 10 and 17 are not.

Mathematica

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 *)

Prog

(Python)
def ROR(n):                # returns A038572(n)
    BL = len(bin(n))-2
    return (n>>1) + ((n&1) << (BL-1))
def ROL(n):                # returns A006257(n)
    BL = len(bin(n))-2
    return (n*2) - (1<<BL) + 1
for n in range(1000):
    X = n
    Xs = []
    while not (X in Xs):
        Xs.append(X)
        if X==0:
            print str(n)+', ',
            break
        X = X ^ ROR(X) ^ ROL(X)

Crossrefs

Cf. A038572, A006257.

Sequence in context: A279702 A014980 A066901 * A019125 A019205 A191472

Adjacent sequences: A273036 A273037 A273038 * A273040 A273041 A273042

Keyword

nonn,base

Author

Alex Ratushnyak, May 13 2016

Status

approved