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

Author

Ruud H.G. van Tol, Mar 16 2025

Keywords

• nonn
• base

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.

Examples

			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.
		

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

Crossrefs

Cf. A030190, A083652.

Programs

PARI

lista(n)= my(b=List(), i=0, s=0, m=Map(Mat([0, 0])), r=vector(n)); while(s
				

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