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