A279125 Lexicographically earliest sequence such that, for any distinct i and j, a(i)=a(j) implies (i AND j)=0 (where AND stands for the bitwise AND operator).

0, 0, 1, 0, 2, 3, 4, 0, 3, 2, 5, 1, 6, 7, 8, 0, 7, 6, 9, 5, 10, 11, 12, 4, 13, 14, 15, 16, 17, 18, 19, 0, 11, 10, 16, 9, 14, 13, 20, 12, 21, 22, 23, 24, 25, 26, 27, 1, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 0, 18, 17, 24, 15, 22, 21, 35, 9

Offset: 1

Author

Rémy Sigrist, Dec 06 2016

Keywords

• nonn
• base
• look

Comments

This sequence is similar to A279119 in the sense that here we check for common ones in binary representation and there we check for common prime factors.

By analogy with A275152, this sequence can be seen as a way to tile the first quadrant with fixed disconnected 2-dimensional polyominoes: the (vertical) polyomino corresponding to n is shifted to the right as little as possible so as not to overlap a previous polyomino, and a(n) gives the corresponding number of steps to the right (see illustration in Links section).

Links

• Rémy Sigrist, Table of n, a(n) for n = 1..10000
• Rémy Sigrist, Illustration of the first terms (by way of polyominos)
• N. J. A. Sloane and Brady Haran, Amazing Graphs III, Numberphile video (2019).

Crossrefs

Cf. A000079, A164346, A275152, A279119.

Programs

Maple

with(Bits):
n:= 100:
l:= []:
g:=[seq(0, i = 0..n-1)]:
for i from 1 to n by 1
do
a:= 0;
while (And(g[a + 1], i)) > 0
do
a++;
end do:
g[a + 1] += i;
l:= [op(l), a];
end do:
print(l); # Reza K Ghazi, Dec 29 2021

Mathematica

n = 100;
l = {};
g = ConstantArray[0, n];
For[i = 0, i < n, i++; a = 0; While[BitAnd[g[[a + 1]], i] > 0, a++];
  g[[a + 1]] += i;
  l = Append[l, a]];
l (* Reza K Ghazi, Dec 29 2021 *)

PARI

g = vector(72); for (n=1, #g, a = 0; while (bitand(g[a+1],n)>0, a++); g[a+1] += n; print1 (a", "))

Python

n = 100
g = n * [0]
for i in range(1, n + 1):
    a = 0
    while g[a] & i:
        a += 1
    g[a] += i
    print(a, end=', ') # Reza K Ghazi, Dec 29 2021

Formula

a(n)=0 iff n belongs to A000079.

a(n)=1 iff n belongs to A164346.