A286262 Numbers whose binary expansion is a cubefree string.

0, 1, 2, 3, 4, 5, 6, 9, 10, 11, 12, 13, 18, 19, 20, 21, 22, 25, 26, 27, 36, 37, 38, 41, 43, 44, 45, 50, 51, 52, 53, 54, 73, 74, 75, 76, 77, 82, 83, 86, 89, 90, 91, 100, 101, 102, 105, 107, 108, 109, 146, 147, 148, 150, 153, 154, 155, 164, 165, 166, 172, 173, 178, 179, 180, 181, 182

Offset: 1

Author

M. F. Hasler, May 05 2017

Keywords

• nonn
• base

Comments

Cubefree means that there is no substring which is the repetition of three identical nonempty strings, see examples.

If n is not in the sequence, no number of the form n*2^k + m with 0 <= m < 2^k can be in the sequence, nor any number of the form m*2^k + n with 2^k > n, m >= 0.

Examples

			7 is not in the sequence, because 7 = 111[2] contains three consecutive "1"s.
8 is not in the sequence, because 8 = 1000[2] contains three consecutive "0"s.
More generally, no number congruent to 7 or congruent to 0 (mod 8) may be in the sequence.
Even more generally, no number of the form m*2^(k+3) +- n, n < 2^k, can be in this sequence.
42 is not in the sequence, because 42 = 101010[2] contains three consecutive "10"s.
From the comment follows that no number of the form 7*2^k, 8*2^k or 42*2^k may be in the sequence, for any k>=0. More generally, no number of the form 7*2^k + m, 8*2^k + m or 42*2^k + m may be in the sequence, for any 2^k > m >= 0.
		

Links

• Chai Wah Wu, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A028445, A063037, A286261 (complement of this sequence).

Programs

Maple

isCubeFree:=proc(v) local n, L;
for n from 3 to nops(v) do for L to n/3 do
if v[n-L*2+1 .. n] = v[n-L*3+1 .. n-L] then RETURN(false) fi od od; true end;
a:=[];
for n from 1 to 512 do
if isCubeFree(convert(n, base, 2)) then a:=[op(a), n]; fi; od;
a;

Python

from _future_ import division
def is_cubefree(s):
    l = len(s)
    for i in range(l-2):
        for j in range(1,(l-i)//3+1):
            if s[i:i+2*j] == s[i+j:i+3*j]:
                return False
    return True
A286262_list = [n for n in range(10**4) if is_cubefree(bin(n)[2:])] # Chai Wah Wu, May 06 2017

Formula

lim a(n)/n = infinity: sequence has asymptotic density 0.