A227288 -id:A227288 - cp's OEIS frontend

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

Reinhard Zumkeller, Jul 01 2013

nonn

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)).

			.   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,..} .

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for sequences that are permutations of the natural numbers

Cf. A227114 (inverse).

Cf. A101369, A184992.

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

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

A227289 Smallest m such that gcd(A227113(m+1), A227113(m)) = n.

Original entry on oeis.org

1, 2, 4, 10, 6, 177, 8, 514, 500, 349, 14, 4791, 16, 5949, 1623, 863, 20, 2629, 22, 5113, 2043, 3211, 28, 7347, 4593, 11911, 1647, 12819, 34, 27325, 36, 15755, 34527, 29319, 5459, 2797, 42, 24535, 20145, 15511, 46, 26523, 48, 64895, 13335, 40769, 54, 43835, 184041

Offset: 1

Reinhard Zumkeller, Jul 05 2013

nonn

A227288(a(n)) = n and A227288(m) <> n for m < a(n).

			.   n |   a(n) | A227288(a(n)) = gcd(A227113(a(n)),A227113(a(n)+1))
.  ---+--------+--------------- -----------------------------------
.   1 |     1  |            1  =  gcd(1,2)
.   2 |     2  |            2  =  gcd(2,4)
.   3 |     4  |            3  =  gcd(3,6)
.   4 |    10  |            4  =  gcd(8,12)
.   5 |     6  |            5  =  gcd(5,10)
.   6 |   177  |            6  =  gcd(168,162)
.   7 |     8  |            7  =  gcd(7,14)
.   8 |   514  |            8  =  gcd(472,480)
.   9 |   500  |            9  =  gcd(459,468)
.  10 |   349  |           10  =  gcd(330,320)
.  11 |    14  |           11  =  gcd(11,22)
.  12 |  4791  |           12  =  gcd(4524,4512)

import Data.List (elemIndex); import Data.Maybe (fromJust)
a227289 = (+ 1) . fromJust . (`elemIndex` a227288_list)

cp's OEIS Frontend

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

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