cp's OEIS Frontend

A319268 Lexicographically earliest sequence of distinct positive terms such that for any n > 0, the binary representation of n^2 starts with the binary representation of a(n).

%I A319268 #15 Nov 09 2018 07:33:45
%S A319268 1,2,4,8,3,9,6,16,5,12,7,18,10,24,14,32,36,20,11,25,13,15,33,72,19,21,
%T A319268 22,49,26,28,30,64,17,144,38,40,42,45,23,50,52,27,57,60,31,66,34,288,
%U A319268 37,39,81,84,43,91,47,98,101,105,54,56,29,120,62,128,132,68
%N A319268 Lexicographically earliest sequence of distinct positive terms such that for any n > 0, the binary representation of n^2 starts with the binary representation of a(n).
%C A319268 This sequence is a permutation of the natural numbers with inverse A319499.
%C A319268 We can build a variant of this sequence for any base b > 1.
%C A319268 We can build a variant of this sequence for any strictly increasing sequence of nonnegative integers.
%H A319268 Ivan Neretin, <a href="/A319268/b319268.txt">Table of n, a(n) for n = 1..10000</a>
%H A319268 Rémy Sigrist, <a href="/A319268/a319268.gp.txt">PARI program for A319268</a>
%H A319268 Rémy Sigrist, <a href="/A319268/a319268.png">Colored logarithmic scatterplot of the first 10000 terms</a> (where the color is function of A070939(n^2) - A070939(a(n)))
%H A319268 <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%e A319268 The first terms, alongside the binary representation of n^2 with a(n) in parentheses, are:
%e A319268   n   a(n)  bin(n^2)
%e A319268   --  ----  --------
%e A319268    1     1          (1)
%e A319268    2     2        (10)0
%e A319268    3     4       (100)1
%e A319268    4     8      (1000)0
%e A319268    5     3      (11)001
%e A319268    6     9     (1001)00
%e A319268    7     6     (110)001
%e A319268    8    16    (10000)00
%e A319268    9     5    (101)0001
%e A319268   10    12    (1100)100
%e A319268   11     7    (111)1001
%e A319268   12    18   (10010)000
%e A319268   13    10   (1010)1001
%e A319268   14    24   (11000)100
%e A319268   15    14   (1110)0001
%e A319268   16    32  (100000)000
%e A319268   17    36  (100100)001
%e A319268   18    20  (10100)0100
%e A319268   19    11  (1011)01001
%e A319268   20    25  (11001)0000
%t A319268 a = {1}; Do[r = IntegerDigits[n^2, 2]; AppendTo[a, Min@Complement[Table[FromDigits[Take[r, k], 2], {k, Length@r}],a]], {n, 2, 66}]; a (* _Ivan Neretin_, Oct 24 2018 *)
%o A319268 (PARI) See Links section.
%Y A319268 Cf. A000290, A070939, A272679, A319499.
%K A319268 nonn,base
%O A319268 1,2
%A A319268 _Rémy Sigrist_, Sep 16 2018