A366027 Lexicographically earliest sequence of distinct positive integers such that for any n > 0, if 2^(d-1) appears in the binary expansion of a(n) then d divides n.

1, 2, 4, 3, 16, 5, 64, 8, 256, 17, 1024, 6, 4096, 65, 20, 9, 65536, 7, 262144, 10, 68, 1025, 4194304, 11, 16777216, 4097, 257, 66, 268435456, 18, 1073741824, 128, 1028, 65537, 80, 12, 68719476736, 262145, 4100, 19, 1099511627776, 32, 4398046511104, 1026, 21

Offset

1,2

Comments

In other words, the binary expansion of a(n) encodes a subset of the divisors of n.

This sequence is a permutation of the positive integers with inverse A366028.

Links

Table of n, a(n) for n=1..45.

Rémy Sigrist, PARI program

Index entries for sequences that are permutations of the natural numbers

Formula

a(p) = 2^(p-1) for any prime number p.

a(2*p) = 2^(p-1) + 1 for any prime number p.

Example

The first terms, alongside their binary expansion and the corresponding divisors d, are:
  n   a(n)    bin(a(n))            Corresponding divisors
  --  ------  -------------------  ----------------------
   1       1                    1  {1}
   2       2                   10  {2}
   3       4                  100  {3}
   4       3                   11  {2, 1}
   5      16                10000  {5}
   6       5                  101  {3, 1}
   7      64              1000000  {7}
   8       8                 1000  {4}
   9     256            100000000  {9}
  10      17                10001  {5, 1}
  11    1024          10000000000  {11}
  12       6                  110  {3, 2}
  13    4096        1000000000000  {13}
  14      65              1000001  {7, 1}
  15      20                10100  {5, 3}
  16       9                 1001  {4, 1}
  17   65536    10000000000000000  {17}
  18       7                  111  {3, 2, 1}

Prog

(PARI) See Links section.

Crossrefs

Cf. A048793, A271410, A366028 (inverse).

Sequence in context: A115399 A109429 A358820 * A114894 A183169 A334866

Adjacent sequences: A366024 A366025 A366026 * A366028 A366029 A366030

Keyword

nonn,base

Author

Rémy Sigrist, Sep 26 2023

Status

approved