cp's OEIS Frontend

A286262 Numbers whose binary expansion is a cubefree string.

%I A286262 #16 May 06 2017 20:46:46
%S A286262 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,
%T A286262 44,45,50,51,52,53,54,73,74,75,76,77,82,83,86,89,90,91,100,101,102,
%U A286262 105,107,108,109,146,147,148,150,153,154,155,164,165,166,172,173,178,179,180,181,182
%N A286262 Numbers whose binary expansion is a cubefree string.
%C A286262 Cubefree means that there is no substring which is the repetition of three identical nonempty strings, see examples.
%C A286262 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.
%H A286262 Chai Wah Wu, <a href="/A286262/b286262.txt">Table of n, a(n) for n = 1..10000</a>
%F A286262 lim a(n)/n = infinity: sequence has asymptotic density 0.
%e A286262 7 is not in the sequence, because 7 = 111[2] contains three consecutive "1"s.
%e A286262 8 is not in the sequence, because 8 = 1000[2] contains three consecutive "0"s.
%e A286262 More generally, no number congruent to 7 or congruent to 0 (mod 8) may be in the sequence.
%e A286262 Even more generally, no number of the form m*2^(k+3) +- n, n < 2^k, can be in this sequence.
%e A286262 42 is not in the sequence, because 42 = 101010[2] contains three consecutive "10"s.
%e A286262 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.
%p A286262 isCubeFree:=proc(v) local n, L;
%p A286262 for n from 3 to nops(v) do for L to n/3 do
%p A286262 if v[n-L*2+1 .. n] = v[n-L*3+1 .. n-L] then RETURN(false) fi od od; true end;
%p A286262 a:=[];
%p A286262 for n from 1 to 512 do
%p A286262 if isCubeFree(convert(n, base, 2)) then a:=[op(a), n]; fi; od;
%p A286262 a;
%o A286262 (Python)
%o A286262 from __future__ import division
%o A286262 def is_cubefree(s):
%o A286262     l = len(s)
%o A286262     for i in range(l-2):
%o A286262         for j in range(1,(l-i)//3+1):
%o A286262             if s[i:i+2*j] == s[i+j:i+3*j]:
%o A286262                 return False
%o A286262     return True
%o A286262 A286262_list = [n for n in range(10**4) if is_cubefree(bin(n)[2:])] # _Chai Wah Wu_, May 06 2017
%Y A286262 Cf. A028445, A063037, A286261 (complement of this sequence).
%K A286262 nonn,base
%O A286262 1,3
%A A286262 _M. F. Hasler_, May 05 2017