A227113 Lexicographically earliest permutation of the natural numbers such that all pairs of even- and odd-indexed terms have a common divisor > 1.

1, 2, 4, 3, 6, 5, 10, 7, 14, 8, 12, 9, 15, 11, 22, 13, 26, 16, 18, 17, 34, 19, 38, 20, 24, 21, 27, 23, 46, 25, 30, 28, 32, 29, 58, 31, 62, 33, 36, 35, 40, 37, 74, 39, 42, 41, 82, 43, 86, 44, 48, 45, 50, 47, 94, 49, 56, 51, 54, 52, 60, 53, 106, 55, 65, 57, 63

Offset

1,2

Comments

a(2*n) = smallest number not occurring earlier;

a(2*n+1) = smallest number having with a(2*n) a common divisor greater than 1 and not occurring earlier;

A227288(n) = gcd(a(n), a(n+1)).

Links

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Index entries for sequences that are permutations of the natural numbers

Example

.   n | a(2n) a(2n+1) | GCD |         not occurring after step n
.  ---+---------------+-----+-------------------------------------------
.   0 |    _      1   |   _ |  {2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,..}
.   1 |    2      4   |   2 |  {3,5,6,7,8,9,10,11,12,13,14,15,16,17,..}
.   2 |    3      6   |   3 |  {5,7,8,9,10,11,12,13,14,15,16,17,18,19,..}
.   3 |    5     10   |   5 |  {7,8,9,11,12,13,14,15,16,17,18,19,20,..}
.   4 |    7     14   |   7 |  {8,9,11,12,13,15,16,17,18,19,20,21,22,..}
.   5 |    8     12   |   4 |  {9,11,13,15,16,17,18,19,20,21,22,23,24..}
.   6 |    9     15   |   3 |  {11,13,16,17,18,19,20,21,22,23,24,25,..}
.   7 |   11     22   |  11 |  {13,16,17,18,19,20,21,23,24,25,26,27,..}
.   8 |   13     26   |  11 |  {16,17,18,19,20,21,23,24,25,27,28,29,..}
.   9 |   16     18   |   2 |  {17,19,20,21,23,24,25,27,28,29,30,31,..} .

Prog

(Haskell)
import Data.List (delete)
a227113 n = a227113_list !! (n-1)
a227113_list = 1 : f [2..] where
   f (x:xs) = x : y : f (delete y xs)
     where y : _ = filter ((> 1) . (gcd x)) xs

Crossrefs

Cf. A227114 (inverse).

Cf. A101369, A184992.

Sequence in context: A127202 A179869 A179881 * A034701 A375563 A091857

Adjacent sequences: A227110 A227111 A227112 * A227114 A227115 A227116

Keyword

nonn

Author

Reinhard Zumkeller, Jul 01 2013

Extensions

Thanks to Zak Seidov (who suggested more elaboration) from Reinhard Zumkeller, Jul 05 2013

Status

approved