cp's OEIS Frontend

A320441 Numbers whose binary expansion is quasiperiodic.

%I A320441 #27 Oct 04 2023 17:22:31
%S A320441 3,7,10,15,21,31,36,42,45,54,63,73,85,91,109,127,136,146,153,170,173,
%T A320441 181,182,187,204,219,221,238,255,273,292,307,341,365,375,409,438,443,
%U A320441 477,511,528,546,561,585,594,614,627,660,682,685,693,725,726,731,750
%N A320441 Numbers whose binary expansion is quasiperiodic.
%C A320441 The binary representation of a term (ignoring leading zeros) can be covered by (possibly overlapping) occurrences of one of its proper prefix.
%C A320441 This sequence contains A121016.
%C A320441 For any k > 0, there are A320434(k)/2 terms with binary length k.
%H A320441 Michael S. Branicky, <a href="/A320441/b320441.txt">Table of n, a(n) for n = 1..10000</a>
%H A320441 Rémy Sigrist, <a href="/A320441/a320441.png">Scatterplot of the first difference of the first 100000 terms</a>
%F A320441 A020330(a(n)) belongs to the sequence for any n > 0.
%F A320441 A297405(a(n)) belongs to the sequence for any n > 0.
%e A320441 The first terms, alongside their binary representations and prefixes, are:
%e A320441   n   a(n)  bin(a(n))  prefix
%e A320441   --  ----  ---------  ------
%e A320441    1     3         11       1
%e A320441    2     7        111       1
%e A320441    3    10       1010      10
%e A320441    4    15       1111       1
%e A320441    5    21      10101     101
%e A320441    6    31      11111       1
%e A320441    7    36     100100     100
%e A320441    8    42     101010      10
%e A320441    9    45     101101     101
%e A320441   10    54     110110     110
%e A320441   11    63     111111       1
%e A320441   12    73    1001001    1001
%o A320441 (PARI) isok(w) = { my (tt=0); for (l=1, oo, my (t=w%(2^l)); if (t!=tt, if (t==w, return (0)); my (r=w, g=l); while (g-->=0 && r>=t, r \= 2; if (r%(2^l)==t, if (r==t, return (1), g=l))); tt = t)) }
%o A320441 (Python)
%o A320441 def qp(w):
%o A320441     for i in range(1, len(w)):
%o A320441         prefix, covered = w[:i], set()
%o A320441         for j in range(len(w)-i+1):
%o A320441             if w[j:j+i] == prefix:
%o A320441                 covered |= set(range(j, j+i))
%o A320441         if covered == set(range(len(w))):
%o A320441             return True
%o A320441     return False
%o A320441 def ok(n): return qp(bin(n)[2:])
%o A320441 print([k for k in range(751) if ok(k)]) # _Michael S. Branicky_, Mar 20 2022
%Y A320441 Cf. A020330, A121016, A297405, A320434.
%K A320441 nonn,base
%O A320441 1,1
%A A320441 _Rémy Sigrist_, Jan 09 2019