A295029 Lexicographically earliest sequence of distinct positive terms such that, for any n > 1 with binary expansion (b_1, b_2, ..., b_k) (where b_1 = 1 is the most significant bit of n), a(n) is a multiple of a(i) for each i such that b_i = 1.

1, 2, 4, 3, 8, 6, 12, 5, 9, 16, 24, 10, 18, 20, 36, 7, 32, 15, 48, 28, 40, 60, 72, 14, 56, 30, 96, 44, 64, 84, 120, 11, 42, 80, 144, 21, 54, 168, 192, 52, 108, 88, 216, 132, 156, 240, 264, 22, 66, 104, 288, 78, 90, 312, 336, 68, 180, 112, 360, 204, 228, 384

Offset

1,2

Comments

This sequence is a permutation of the natural numbers, with inverse A297499; as a(1) = 1, for any k > 1, a(2^k) has only to be a multiple of 1, and so a(2^k) will be the least unused value, and eventually any number will appear in the sequence.

Prime numbers can only appear at positions that are powers of 2.

For any n > 1, a(n) is a multiple of m(n) = lcm(a(e_1), ..., a(e_h)) where the list (e_1, ..., e_h) corresponds to the ones in the binary expansion of n (in particular, e_1 = 1 and h = A000120(n)); the lines and dashed lines visible in the logarithmic scatterplot of the first terms correspond to sets of terms a(n) where m(n) has the same value (see Links section).

Links

Rémy Sigrist, Table of n, a(n) for n = 1..50000

Index entries for sequences that are permutations of the natural numbers

Rémy Sigrist, Colored logarithmic scatterplot of the first 150000 terms (where the color is function of m(n))

Rémy Sigrist, PARI program for A295029

Example

The first terms, alongside the binary expansion of n and m(n), are:
  n     a(n)    bin(n)    m(n)
  --    ----    ------    -------------
   1       1         1     1 = lcm(a(1))
   2       2        10     1 = lcm(a(1))
   3       4        11     2 = lcm(a(1), a(2))
   4       3       100     1 = lcm(a(1))
   5       8       101     4 = lcm(a(1), a(3))
   6       6       110     2 = lcm(a(1), a(2))
   7      12       111     4 = lcm(a(1), a(2), a(3))
   8       5      1000     1 = lcm(a(1))
   9       9      1001     3 = lcm(a(1), a(4))
  10      16      1010     4 = lcm(a(1), a(3))
  11      24      1011    12 = lcm(a(1), a(3), a(4))
  12      10      1100     2 = lcm(a(1), a(2))
  13      18      1101     6 = lcm(a(1), a(2), a(4))
  14      20      1110     4 = lcm(a(1), a(2), a(3))
  15      36      1111    12 = lcm(a(1), a(2), a(3), a(4))
  16       7     10000     1 = lcm(a(1))
  17      32     10001     8 = lcm(a(1), a(5))
  18      15     10010     3 = lcm(a(1), a(4))
  19      48     10011    24 = lcm(a(1), a(4), a(5))
  20      28     10100     4 = lcm(a(1), a(3))

Prog

(PARI) See Links section.

Crossrefs

Cf. A000120, A297499 (inverse).

Sequence in context: A294044 A243072 A243346 * A338918 A329605 A243073

Adjacent sequences: A295026 A295027 A295028 * A295030 A295031 A295032

Keyword

nonn,base

Author

Rémy Sigrist, Dec 30 2017

Status

approved