A370727 Lexicographically earliest sequence of distinct positive integers such that for any n > 0, prime(n) AND a(n) = a(n) (where prime(n) denotes the n-th prime number and AND denotes the bitwise AND operator).

2, 1, 4, 3, 8, 5, 16, 17, 6, 9, 7, 32, 33, 10, 11, 20, 18, 12, 64, 65, 72, 13, 19, 24, 96, 36, 34, 35, 37, 48, 14, 128, 129, 130, 21, 22, 25, 131, 38, 40, 49, 52, 15, 192, 68, 66, 67, 23, 97, 69, 41, 39, 80, 26, 256, 257, 260, 258, 261, 264, 27, 288, 50, 51

Offset

1,1

Comments

In other words, the 1's in the binary expansion of the n-th term also appear in that of the n-th prime number.

This sequence is a permutation of the positive integers with inverse A370727: for any w > 0, there are infinitely many prime numbers whose binary expansions end with w 1's, and these are all occasions for an integer < 2^w to appear in the sequence.

Links

Rémy Sigrist, Table of n, a(n) for n = 1..10000

Rémy Sigrist, PARI program

Index entries for sequences that are permutations of the natural numbers

Example

The first terms, alongside the corresponding binary expansions, are:
  n   a(n)  bin(a(n))  bin(prime(n))
  --  ----  ---------  -------------
   1     2         10             10
   2     1          1             11
   3     4        100            101
   4     3         11            111
   5     8       1000           1011
   6     5        101           1101
   7    16      10000          10001
   8    17      10001          10011
   9     6        110          10111
  10     9       1001          11101

Prog

(PARI) See Links section.

Crossrefs

Cf. A295609, A295989, A370728 (inverse).

Sequence in context: A275902 A008347 A112387 * A193174 A126084 A294022

Adjacent sequences: A370724 A370725 A370726 * A370728 A370729 A370730

Keyword

nonn,base

Author

Rémy Sigrist, Feb 28 2024

Status

approved