cp's OEIS Frontend

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.

%I A382115 #34 Mar 27 2025 20:28:32
%S A382115 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,
%T A382115 19,38,77,10,21,42,85,43,87,47,94,28,25,51,102,205,155,311,111,222,29,
%U A382115 59,119,239,31,62,60,24,16,33,66,132,264,529,35,70,140,281,50
%N 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.
%C A382115 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.
%C A382115 This sequence is a permutation of the natural numbers.
%H A382115 Ruud H.G. van Tol, <a href="/A382115/b382115.txt">Table of n, a(n) for n = 1..10000</a>
%H A382115 <a href="/index/Per#IntegerPermutation">Index entries for sequences that are a permutation of the natural numbers</a>
%H A382115 <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>
%e A382115 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.
%e A382115 Juxtaposed numbers in binary, and positions within them, begin
%e A382115   n = 1  2 3  4 5  6 7 8  ...
%e A382115       1  1 0  1 1  1 0 0  ...
%e A382115                 \----/
%e A382115 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.
%o A382115 (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;
%o A382115 (Python)
%o A382115 from itertools import count, islice
%o A382115 def bgen(): # generates concatenation of binary of digits of 1..oo
%o A382115     yield from (b for i in count(1) for b in bin(i)[2:])
%o A382115 def agen(): # generator of terms
%o A382115     aset, s, g = set(), "", bgen()
%o A382115     for n in count(1):
%o A382115         s += next(g)
%o A382115         an = next(i for k in range(1, n+1) if (i:=int(s[-k:], 2)) not in aset and i > 0)
%o A382115         yield an
%o A382115         aset.add(an)
%o A382115 print(list(islice(agen(), 64))) # _Michael S. Branicky_, Mar 16 2025
%Y A382115 Cf. A030190, A083652.
%K A382115 nonn,base
%O A382115 1,2
%A A382115 _Ruud H.G. van Tol_, Mar 16 2025