A327119 Sequence obtained by swapping each (k*(2n))-th element of the nonnegative integers with the (k*(2n+1))-th element, for all k>0 in ascending order, omitting the first term.

0, 1, 3, 2, 7, 4, 8, 6, 14, 5, 15, 10, 20, 12, 17, 9, 34, 16, 27, 18, 31, 13, 29, 22, 47, 19, 39, 11, 48, 28, 44, 30, 76, 21, 51, 26, 62, 36, 53, 25, 69, 40, 55, 42, 75, 24, 65, 46, 97, 35, 63, 33, 94, 52, 71, 43, 95, 37, 87, 58, 90, 60, 89, 32, 167, 50, 84

Offset

1,3

Comments

The first term must be omitted because it does not converge.

Start with the sequence of nonnegative integers [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, ...].

Swap all pairs specified by k=1, resulting in [1, 0, 3, 2, 5, 4, 7, 6, 9, 8, 11, ...], so the first term of the final sequence is 0 (No swaps for k>1 will affect this term).

Swap all pairs specified by k=2, resulting in [3, 0, 1, 2, 7, 4, 5, 6, 11, 8, 9, ...], so the second term of the final sequence is 1 (No swaps for k>2 will affect this term).

Swap all pairs specified by k=3, resulting in [2, 0, 1, 3, 7, 4, 8, 6, 11, 5, 9, ...], so the third term of the final sequence is 3 (No swaps for k>3 will affect this term).

Continue for all values of k.

a(n) is equivalent to -A327093(-n), if A327093 is extended to all integers.

It appears that n is an odd prime number iff a(n+1)=n-1. If true, is there a formal analogy with the Sieve of Eratosthenes (by swapping instead of marking terms), or is this another type of sieve? - Jon Maiga, May 31 2021

Links

Jennifer Buckley, Table of n, a(n) for n = 1..10000

Formula

a(n) = A004442(A327420(n)) (conjectured). - Jon Maiga, May 31 2021

Prog

(Go)
func a(n int) int {
    for k := n; k > 0; k-- {
        if n%k == 0 {
            if (n/k)%2 == 0 {
                n = n + k
            } else {
                n = n - k
            }
        }
    }
    return n
}

Crossrefs

Cf. A004442, A327093, A327420.

Inverse: A327120.

Sequence in context: A175055 A351207 A085168 * A113658 A318462 A273926

Adjacent sequences: A327116 A327117 A327118 * A327120 A327121 A327122

Keyword

nonn

Author

Jennifer Buckley, Sep 13 2019

Status

approved