A327093 - OEIS

%I #61 Jul 17 2023 15:17:36

%S 2,3,7,5,11,13,15,10,17,19,23,25,27,21,40,16,35,36,39,37,58,33,47,50,

%T 52,43,45,34,59,78,63,31,76,55,82,67,75,57,99,56,83,112,87,61,126,69,

%U 95,92,97,96,133,71,107,81,142,79,139,91,119,155,123,93,122,51,151,146,135

%N Sequence obtained by swapping each (k*(2n))-th element of the positive integers with the (k*(2n-1))-th element, for all k > 0, in ascending order.

%C Start with the sequence of positive integers [1, 2, 3, 4, 5, 6, 7, 8, ...].

%C Swap all pairs specified by k=1, that is, do the swaps (2,1),(4,3),(6,5),(8,7),..., resulting in [2, 1, 4, 3, 6, 5, 8, 7, ...], so the first term of the final sequence is 2 (No swaps for k>1 will affect this term).

%C Swap all pairs specified by k=2, that is, do the swaps (4,2),(8,6),(12,10),(16,14),..., resulting in [2, 3, 4, 1, 6, 7, 8, 5, ...], so the second term of the final sequence is 3 (No swaps for k>2 will affect this term).

%C Swap all pairs specified by k=3, that is, do the swaps (6,3),(12,9),(18,15),(24,21),... .

%C Continue for all values of k.

%C The complementary sequence 1, 4, 6, 8, 9, 12, 14, 18, 20, 22, 24, 26, 28, ... lists the numbers that never appear. Is there an alternative characterization of these numbers?

%C Equivalently, is there a characterization of the numbers (2, 3, 5, 7, 10, 11, 13, 15, 16, 17, 19, 21, 23, ...) that do appear? - _N. J. A. Sloane_, Sep 13 2019

%H Jennifer Buckley, <a href="/A327093/b327093.txt">Table of n, a(n) for n = 1..10000</a>

%o (Go)

%o func a(n int) int {

%o     for k := n; k > 0; k-- {

%o         if n%k == 0 {

%o             if (n/k)%2 == 0 {

%o                 n = n - k

%o             } else {

%o                 n = n + k

%o             }

%o         }

%o     }

%o     return n

%o }

%o (SageMath)

%o def a(n):

%o     for k in srange(n, 0, -1):

%o         if k.divides(n):

%o             n += k if is_odd(n//k) else -k

%o     return n

%o print([a(n) for n in (1..67)]) # _Peter Luschny_, Sep 14 2019

%Y For the sorted terms and the missing terms see A327445, A327446.

%Y Cf. A327119, A064494, A064627, A057032, A069829, A327420.

%K nonn

%O 1,1

%A _Jennifer Buckley_, Sep 13 2019