A382115 a(n) is the smallest positive number not already used and whose binary expansion occurs, ending at position n, in the binary Champernowne word.

1, 3, 2, 5, 11, 7, 6, 4, 9, 18, 37, 75, 23, 14, 13, 27, 55, 15, 30, 12, 8, 17, 34, 68, 137, 19, 38, 77, 10, 21, 42, 85, 43, 87, 47, 94, 28, 25, 51, 102, 205, 155, 311, 111, 222, 29, 59, 119, 239, 31, 62, 60, 24, 16, 33, 66, 132, 264, 529, 35, 70, 140, 281, 50

Offset

1,2

Comments

In other words, distinct positive numbers in base-10 from the digits n-k to n from the juxtaposition of the positive binary numbers, with k >= 0 minimal.

This sequence is a permutation of the natural numbers.

Links

Ruud H.G. van Tol, Table of n, a(n) for n = 1..10000

Index entries for sequences that are a permutation of the natural numbers

Index entries for sequences related to binary expansion of n

Example

From the juxtaposition 1.10.11.100.101.110.111.1000..., a(1) uses digits 1 to 1, a(2) uses digits 1 to 2, a(3) uses digits 2 to 3 and a(4) uses digits 2 to 4.
Juxtaposed numbers in binary, and positions within them, begin
  n = 1  2 3  4 5  6 7 8  ...
      1  1 0  1 1  1 0 0  ...
                \----/
For n=7, binary 10 = 2 ends at n=7 but has already appeared at a(3)=2, and the next smallest binary 110 = 6 has not yet appeared so a(7) = 6.

Prog

(PARI) lista(n)= my(b=List(), i=0, s=0, m=Map(Mat([0, 0])), r=vector(n)); while(s<n, listput(~b, binary(i++)); s+=#b[#b]); b=concat(b); for(j=1, n, forstep(i=j, 1, -1, my(a=fromdigits(b[i..j], 2)); if(!mapisdefined(m, a), r[j]=a; mapput(~m, a, 0); break))); r;
(Python)
from itertools import count, islice
def bgen(): # generates concatenation of binary of digits of 1..oo
    yield from (b for i in count(1) for b in bin(i)[2:])
def agen(): # generator of terms
    aset, s, g = set(), "", bgen()
    for n in count(1):
        s += next(g)
        an = next(i for k in range(1, n+1) if (i:=int(s[-k:], 2)) not in aset and i > 0)
        yield an
        aset.add(an)
print(list(islice(agen(), 64))) # Michael S. Branicky, Mar 16 2025

Crossrefs

Cf. A030190, A083652.

Sequence in context: A257905 A379657 A305878 * A093924 A130597 A387365

Adjacent sequences: A382112 A382113 A382114 * A382116 A382117 A382118

Keyword

nonn,base

Author

Ruud H.G. van Tol, Mar 16 2025

Status

approved