A289813 A binary encoding of the ones in ternary representation of n (see Comments for precise definition).

0, 1, 0, 2, 3, 2, 0, 1, 0, 4, 5, 4, 6, 7, 6, 4, 5, 4, 0, 1, 0, 2, 3, 2, 0, 1, 0, 8, 9, 8, 10, 11, 10, 8, 9, 8, 12, 13, 12, 14, 15, 14, 12, 13, 12, 8, 9, 8, 10, 11, 10, 8, 9, 8, 0, 1, 0, 2, 3, 2, 0, 1, 0, 4, 5, 4, 6, 7, 6, 4, 5, 4, 0, 1, 0, 2, 3, 2, 0, 1, 0, 16

Offset: 0

Author

Rémy Sigrist, Jul 12 2017

Keywords

• nonn
• base
• look

Comments

The ones in the binary representation of a(n) correspond to the ones in the ternary representation of n; for example: ternary(42) = 1120 and binary(a(42)) = 1100 (a(42) = 12).

See A289814 for the sequence encoding the twos in ternary representation of n.

By design, a(n) AND A289814(n) = 0 (where AND stands for the bitwise AND operator).

See A289831 for the sum of this sequence and A289814.

For each pair of numbers without common bits in base 2 representation, say x and y, there is a unique index, say n, such that a(n) = x and A289814(n) = y; in fact, n = A289869(x,y).

The scatterplot of this sequence vs A289814 looks like a Sierpinski triangle pivoted to the side.

For any t > 0: we can adapt the algorithm used here and in A289814 in order to uniquely enumerate every tuple of t numbers mutually without common bits in base 2 representation.

Examples

			The first values, alongside the ternary representation of n, and the binary representation of a(n), are:
n       a(n)    ternary(n)  binary(a(n))
--      ----    ----------  ------------
0       0       0           0
1       1       1           1
2       0       2           0
3       2       10          10
4       3       11          11
5       2       12          10
6       0       20          0
7       1       21          1
8       0       22          0
9       4       100         100
10      5       101         101
11      4       102         100
12      6       110         110
13      7       111         111
14      6       112         110
15      4       120         100
16      5       121         101
17      4       122         100
18      0       200         0
19      1       201         1
20      0       202         0
21      2       210         10
22      3       211         11
23      2       212         10
24      0       220         0
25      1       221         1
26      0       222         0
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 0..6560

Crossrefs

Cf. A000120, A004488, A005836, A053735, A289814, A289831, A289869.

Programs

Mathematica

Table[FromDigits[#, 2] &[IntegerDigits[n, 3] /. 2 -> 0], {n, 0, 81}] (* Michael De Vlieger, Jul 20 2017 *)

PARI

a(n) = my (d=digits(n,3)); fromdigits(vector(#d, i, if (d[i]==1, 1, 0)), 2)

PARI

a(n) = fromdigits(digits(n, 3)%2, 2); \\ Ruud H.G. van Tol, May 08 2024

Python

from sympy.ntheory.factor_ import digits
def a(n):
    d = digits(n, 3)[1:]
    return int("".join('1' if i==1 else '0' for i in d), 2)
print([a(n) for n in range(51)]) # Indranil Ghosh, Jul 20 2017

Formula

a(0) = 0.

a(3*n) = 2 * a(n).

a(3*n+1) = 2 * a(n) + 1.

a(3*n+2) = 2 * a(n).

Also, a(n) = A289814(A004488(n)).

A053735(n) = A000120(a(n)) + 2*A000120(A289814(n)). - Antti Karttunen, Jul 20 2017