A286261 Numbers whose binary expansion is not a cubefree string.

7, 8, 14, 15, 16, 17, 23, 24, 28, 29, 30, 31, 32, 33, 34, 35, 39, 40, 42, 46, 47, 48, 49, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 78, 79, 80, 81, 84, 85, 87, 88, 92, 93, 94, 95, 96, 97, 98, 99, 103, 104, 106, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130

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 in the sequence, any number of the form n*2^k + m with 0 <= m < 2^k is in the sequence, and also any number of the form m*2^k + n with 2^k > n, m >= 0.

Examples

			7 is in the sequence, because 7 = 111[2] contains three consecutive "1"s.
8 is in the sequence, because 8 = 1000[2] contains three consecutive "0"s.
42 is in the sequence, because 42 = 101010[2] contains three consecutive "10"s.
From the comment follows that all numbers of the form 7*2^k, 8*2^k or 42*2^k are in the sequence, for any k >= 0.
All numbers congruent to 7 or congruent to 0 (mod 8) are in the sequence.
All numbers of the form m*2^(k+3) +- n with n < 2^k are in the sequence.
		

Links

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

Crossrefs

Cf. A028445, A286262 (complement of this sequence).

Programs

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
A286261_list = [n for n in range(10**4) if not is_cubefree(bin(n)[2:])] # Chai Wah Wu, May 06 2017

Formula

a(n) ~ n: the sequence has asymptotic density one.