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A361640 a(0) = 0, a(1) = 1; thereafter let b be the least power of 2 that does not appear in the binary expansions of a(n-2) and a(n-1), then a(n) is the smallest multiple of b that is not yet in the sequence.

%I A361640 #17 Mar 21 2023 09:23:27
%S A361640 0,1,2,4,3,8,12,5,6,16,7,24,32,9,10,20,11,64,28,13,14,48,15,128,80,17,
%T A361640 18,36,19,40,44,21,22,56,23,192,72,25,26,52,27,256,60,29,30,96,31,384,
%U A361640 160,33,34,68,35,88,76,37,38,104,39,112,120,41,42,84,43
%N A361640 a(0) = 0, a(1) = 1; thereafter let b be the least power of 2 that does not appear in the binary expansions of a(n-2) and a(n-1), then a(n) is the smallest multiple of b that is not yet in the sequence.
%C A361640 This sequence is a variant of A359804; here we consider binary expansions, there prime factorizations.
%C A361640 All powers of 2 appear in the sequence, in ascending order.
%C A361640 This sequence is a permutation of the nonnegative integers (with inverse A361641): an odd term is always followed by two even terms, and after two even terms we can choose the least value not yet in the sequence.
%H A361640 Rémy Sigrist, <a href="/A361640/b361640.txt">Table of n, a(n) for n = 0..8192</a>
%H A361640 Rémy Sigrist, <a href="/A361640/a361640.gp.txt">PARI program</a>
%H A361640 Michael De Vlieger, <a href="/A361640/a361640.png">Log log scatterplot of a(n)</a>, n = 1..2^20.
%H A361640 Michael De Vlieger, <a href="/A361640/a361640_1.png">Log log scatterplot of a(n)</a>, n = 1..2^14, showing primes in red, composite prime powers in gold, squarefree composites in green, and numbers neither squarefree nor prime powers in blue.
%H A361640 <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%e A361640 The first terms, in decimal and in binary, alongside the corresponding b's, are:
%e A361640   n   a(n)  bin(a(n))  b
%e A361640   --  ----  ---------  ---
%e A361640    0     0          0  N/A
%e A361640    1     1          1  N/A
%e A361640    2     2         10    2
%e A361640    3     4        100    4
%e A361640    4     3         11    1
%e A361640    5     8       1000    8
%e A361640    6    12       1100    4
%e A361640    7     5        101    1
%e A361640    8     6        110    2
%e A361640    9    16      10000    8
%e A361640   10     7        111    1
%e A361640   11    24      11000    8
%e A361640   12    32     100000   32
%t A361640 nn = 120; c[_] = False; q[_] = 1;
%t A361640 f[n_] := f[n] = -1 + Position[Reverse@ IntegerDigits[n, 2], 1][[All, 1]];
%t A361640 a[1] = 0; a[2] = 1; c[0] = c[1] = True; i = f[0]; j = f[1];
%t A361640 Do[(k = q[#]; While[c[k #], k++]; q[#] = k; k *= #) &[
%t A361640    2^First@ Complement[Range[0, Max[#] + 1], #] &[Union[i, j]]];
%t A361640   Set[{a[n], c[k], i, j}, {k, True, j, f[k]}], {n, 3, nn}];
%t A361640 Array[a, nn] (* _Michael De Vlieger_, Mar 20 2023 *)
%o A361640 (PARI) See Links section.
%Y A361640 Cf. A006519, A359804, A361641 (inverse).
%K A361640 nonn,base
%O A361640 0,3
%A A361640 _Rémy Sigrist_, Mar 19 2023