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.

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, Table of n, a(n) for n = 0..8192

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.

Index entries for sequences that are permutations of the natural numbers

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

Adjacent sequences: A361637 A361638 A361639 * A361641 A361642 A361643

Keyword

nonn,base

Author

Rémy Sigrist, Mar 19 2023

Status

approved