A320441 - OEIS

A320441

Numbers whose binary expansion is quasiperiodic.

%I #27 Oct 04 2023 17:22:31

%S 3,7,10,15,21,31,36,42,45,54,63,73,85,91,109,127,136,146,153,170,173,

%T 181,182,187,204,219,221,238,255,273,292,307,341,365,375,409,438,443,

%U 477,511,528,546,561,585,594,614,627,660,682,685,693,725,726,731,750

%N Numbers whose binary expansion is quasiperiodic.

%C The binary representation of a term (ignoring leading zeros) can be covered by (possibly overlapping) occurrences of one of its proper prefix.

%C This sequence contains A121016.

%C For any k > 0, there are A320434(k)/2 terms with binary length k.

%H Michael S. Branicky, <a href="/A320441/b320441.txt">Table of n, a(n) for n = 1..10000</a>

%H Rémy Sigrist, <a href="/A320441/a320441.png">Scatterplot of the first difference of the first 100000 terms</a>

%F A020330(a(n)) belongs to the sequence for any n > 0.

%F A297405(a(n)) belongs to the sequence for any n > 0.

%e The first terms, alongside their binary representations and prefixes, are:

%e n a(n) bin(a(n)) prefix

%e -- ---- --------- ------

%e 1 3 11 1

%e 2 7 111 1

%e 3 10 1010 10

%e 4 15 1111 1

%e 5 21 10101 101

%e 6 31 11111 1

%e 7 36 100100 100

%e 8 42 101010 10

%e 9 45 101101 101

%e 10 54 110110 110

%e 11 63 111111 1

%e 12 73 1001001 1001

%o (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 (Python)

%o def qp(w):

%o for i in range(1, len(w)):

%o prefix, covered = w[:i], set()

%o for j in range(len(w)-i+1):

%o if w[j:j+i] == prefix:

%o covered |= set(range(j, j+i))

%o if covered == set(range(len(w))):

%o return True

%o return False

%o def ok(n): return qp(bin(n)[2:])

%o print([k for k in range(751) if ok(k)]) # _Michael S. Branicky_, Mar 20 2022

%Y Cf. A020330, A121016, A297405, A320434.

%K nonn,base

%O 1,1

%A _Rémy Sigrist_, Jan 09 2019