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A361640
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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.
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2
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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, 18, 36, 19, 40, 44, 21, 22, 56, 23, 192, 72, 25, 26, 52, 27, 256, 60, 29, 30, 96, 31, 384, 160, 33, 34, 68, 35, 88, 76, 37, 38, 104, 39, 112, 120, 41, 42, 84, 43
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OFFSET
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0,3
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COMMENTS
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This sequence is a variant of A359804; here we consider binary expansions, there prime factorizations.
All powers of 2 appear in the sequence, in ascending order.
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.
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LINKS
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Michael De Vlieger, Log log scatterplot of a(n), 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.
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EXAMPLE
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The first terms, in decimal and in binary, alongside the corresponding b's, are:
n a(n) bin(a(n)) b
-- ---- --------- ---
0 0 0 N/A
1 1 1 N/A
2 2 10 2
3 4 100 4
4 3 11 1
5 8 1000 8
6 12 1100 4
7 5 101 1
8 6 110 2
9 16 10000 8
10 7 111 1
11 24 11000 8
12 32 100000 32
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MATHEMATICA
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nn = 120; c[_] = False; q[_] = 1;
f[n_] := f[n] = -1 + Position[Reverse@ IntegerDigits[n, 2], 1][[All, 1]];
a[1] = 0; a[2] = 1; c[0] = c[1] = True; i = f[0]; j = f[1];
Do[(k = q[#]; While[c[k #], k++]; q[#] = k; k *= #) &[
2^First@ Complement[Range[0, Max[#] + 1], #] &[Union[i, j]]];
Set[{a[n], c[k], i, j}, {k, True, j, f[k]}], {n, 3, nn}];
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PROG
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(PARI) See Links section.
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CROSSREFS
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KEYWORD
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nonn,base
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AUTHOR
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STATUS
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approved
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