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URL: https://oeis.org/A361640

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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.
2
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
OFFSET
0,3
COMMENTS
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.
LINKS
Rémy Sigrist, PARI program
Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^20.
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.
EXAMPLE
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
MATHEMATICA
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}];
Array[a, nn] (* Michael De Vlieger, Mar 20 2023 *)
PROG
(PARI) See Links section.
CROSSREFS
Cf. A006519, A359804, A361641 (inverse).
Sequence in context: A347976 A387024 A253722 * A323506 A357988 A302747
KEYWORD
nonn,base
AUTHOR
Rémy Sigrist, Mar 19 2023
STATUS
approved