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

1,3

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.

Links

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

Formula

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

Example

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.

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;

Prog

(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

Crossrefs

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

Sequence in context: A032845 A023776 A063037 * A330029 A201992 A329297

Adjacent sequences: A286259 A286260 A286261 * A286263 A286264 A286265

Keyword

nonn,base

Author

M. F. Hasler, May 05 2017

Status

approved