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

%I A227113 #11 Aug 20 2017 23:19:31
%S A227113 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,
%T A227113 27,23,46,25,30,28,32,29,58,31,62,33,36,35,40,37,74,39,42,41,82,43,86,
%U A227113 44,48,45,50,47,94,49,56,51,54,52,60,53,106,55,65,57,63
%N A227113 Lexicographically earliest permutation of the natural numbers such that all pairs of even- and odd-indexed terms have a common divisor > 1.
%C A227113 a(2*n) = smallest number not occurring earlier;
%C A227113 a(2*n+1) = smallest number having with a(2*n) a common divisor greater than 1 and not occurring earlier;
%C A227113 A227288(n) = gcd(a(n), a(n+1)).
%H A227113 Reinhard Zumkeller, <a href="/A227113/b227113.txt">Table of n, a(n) for n = 1..10000</a>
%H A227113 <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%e A227113 .   n | a(2n) a(2n+1) | GCD |         not occurring after step n
%e A227113 .  ---+---------------+-----+-------------------------------------------
%e A227113 .   0 |    _      1   |   _ |  {2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,..}
%e A227113 .   1 |    2      4   |   2 |  {3,5,6,7,8,9,10,11,12,13,14,15,16,17,..}
%e A227113 .   2 |    3      6   |   3 |  {5,7,8,9,10,11,12,13,14,15,16,17,18,19,..}
%e A227113 .   3 |    5     10   |   5 |  {7,8,9,11,12,13,14,15,16,17,18,19,20,..}
%e A227113 .   4 |    7     14   |   7 |  {8,9,11,12,13,15,16,17,18,19,20,21,22,..}
%e A227113 .   5 |    8     12   |   4 |  {9,11,13,15,16,17,18,19,20,21,22,23,24..}
%e A227113 .   6 |    9     15   |   3 |  {11,13,16,17,18,19,20,21,22,23,24,25,..}
%e A227113 .   7 |   11     22   |  11 |  {13,16,17,18,19,20,21,23,24,25,26,27,..}
%e A227113 .   8 |   13     26   |  11 |  {16,17,18,19,20,21,23,24,25,27,28,29,..}
%e A227113 .   9 |   16     18   |   2 |  {17,19,20,21,23,24,25,27,28,29,30,31,..} .
%o A227113 (Haskell)
%o A227113 import Data.List (delete)
%o A227113 a227113 n = a227113_list !! (n-1)
%o A227113 a227113_list = 1 : f [2..] where
%o A227113    f (x:xs) = x : y : f (delete y xs)
%o A227113      where y : _ = filter ((> 1) . (gcd x)) xs
%Y A227113 Cf. A227114 (inverse).
%Y A227113 Cf. A101369, A184992.
%K A227113 nonn
%O A227113 1,2
%A A227113 _Reinhard Zumkeller_, Jul 01 2013
%E A227113 Thanks to _Zak Seidov_ (who suggested more elaboration) from _Reinhard Zumkeller_, Jul 05 2013