A297405 -id:A297405 - cp's OEIS frontend

A320441 Numbers whose binary expansion is quasiperiodic.

3, 7, 10, 15, 21, 31, 36, 42, 45, 54, 63, 73, 85, 91, 109, 127, 136, 146, 153, 170, 173, 181, 182, 187, 204, 219, 221, 238, 255, 273, 292, 307, 341, 365, 375, 409, 438, 443, 477, 511, 528, 546, 561, 585, 594, 614, 627, 660, 682, 685, 693, 725, 726, 731, 750

Offset: 1

Rémy Sigrist, Jan 09 2019

The binary representation of a term (ignoring leading zeros) can be covered by (possibly overlapping) occurrences of one of its proper prefix.
This sequence contains A121016.
For any k > 0, there are A320434(k)/2 terms with binary length k.

			The first terms, alongside their binary representations and prefixes, are:
  n   a(n)  bin(a(n))  prefix
  --  ----  ---------  ------
   1     3         11       1
   2     7        111       1
   3    10       1010      10
   4    15       1111       1
   5    21      10101     101
   6    31      11111       1
   7    36     100100     100
   8    42     101010      10
   9    45     101101     101
  10    54     110110     110
  11    63     111111       1
  12    73    1001001    1001

Michael S. Branicky, Table of n, a(n) for n = 1..10000
Rémy Sigrist, Scatterplot of the first difference of the first 100000 terms

Cf. A020330, A121016, A297405, A320434.

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)) }

def qp(w):
    for i in range(1, len(w)):
        prefix, covered = w[:i], set()
        for j in range(len(w)-i+1):
            if w[j:j+i] == prefix:
                covered |= set(range(j, j+i))
        if covered == set(range(len(w))):
            return True
    return False
def ok(n): return qp(bin(n)[2:])
print([k for k in range(751) if ok(k)]) # Michael S. Branicky, Mar 20 2022

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

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

A337223 a(n) is the least number that can be obtained by replacing some cube XXX in the binary expansion of n by X.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 1, 2, 9, 10, 11, 12, 13, 2, 3, 4, 5, 18, 19, 20, 21, 22, 5, 6, 25, 26, 27, 4, 5, 6, 7, 8, 9, 10, 11, 36, 37, 38, 9, 10, 41, 2, 43, 44, 45, 10, 11, 12, 13, 50, 51, 52, 53, 54, 13, 8, 9, 10, 11, 12, 13, 14, 3, 4, 17, 18, 19, 20, 21, 22, 17

Offset: 0

Rémy Sigrist, Aug 19 2020

Leading zeros in binary expansions are ignored.

Fixed points correspond to A286262.

			The first terms, in decimal and in binary, are:
  n   a(n)  bin(n)  bin(a(n))
  --  ----  ------  ---------
   0     0       0          0
   1     1       1          1
   2     2      10         10
   3     3      11         11
   4     4     100        100
   5     5     101        101
   6     6     110        110
   7     1     111          1
   8     2    1000         10
   9     9    1001       1001
  10    10    1010       1010
  11    11    1011       1011
  12    12    1100       1100
  13    13    1101       1101
  14     2    1110         10
  15     3    1111         11
  16     4   10000        100

Rémy Sigrist, Table of n, a(n) for n = 0..8192
Rémy Sigrist, PARI program for A337223
Index entries for sequences related to binary expansion of n

Cf. A286262, A297405, A337222, A337224.

PARI
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See Links section.
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a(A297405(n)) = n for any n > 0.

cp's OEIS Frontend

A320441 Numbers whose binary expansion is quasiperiodic.

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