Jennifer Buckley - cp's OEIS frontend

A372297 Limit of the recursion B(k) = T[k](B(k-1)), where B(1) = (1,2,3,4,5,...) and T[k] is the transformation that permutes the entries k(2i-1) and k(2i) for all positive integers i, if k is prime.

1, 4, 8, 2, 12, 3, 16, 6, 10, 5, 24, 9, 28, 7, 18, 14, 36, 15, 40, 20, 26, 11, 48, 21, 27, 13, 32, 22, 60, 25, 64, 30, 42, 17, 39, 33, 76, 19, 50, 35, 84, 38, 88, 34, 52, 23, 96, 45, 54, 46, 66, 44, 108, 51, 63, 49, 74, 29, 120, 55, 124, 31, 65, 62, 75

Offset: 1

Jennifer Buckley, Apr 25 2024

nonn

Sequence contains all positive integers.

a(2p) = p for all prime numbers p.

			B(1) = 1,2,3,4, 5,6,7,8, 9,10,11,12,13,14,...
B(2) = 1,4,3,2, 5,8,7,6, 9,12,11,10,13,16,...
B(3) = 1,4,8,2, 5,3,7,6,10,12,11, 9,13,16,...
B(4) = 1,4,8,2, 5,3,7,6,10,12,11, 9,13,16,... (No change)
B(5) = 1,4,8,2,12,3,7,6,10, 5,11, 9,13,16,...

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

Cf. A064494.

max = 66; b[1, j_] := j; b[k_, j_] := b[k, j] = b[k-1, j]; Do[If[PrimeQ[k],b[k, 2j*k-k] = b[k-1, 2j*k]; b[k, 2j*k] = b[k-1, 2j*k-k],b[k,j ]=b[k-1,j]], {k, 2, max}, {j, 1, max}]; a[n_] := b[max, n]; Table[a[n], {n, 1, max}]

A327120 Inverse permutation to A327119.

Original entry on oeis.org

1, 2, 4, 3, 6, 10, 8, 5, 7, 16, 12, 28, 14, 22, 9, 11, 18, 15, 20, 26, 13, 34, 24, 82, 46, 40, 36, 19, 30, 23, 32, 21, 64, 52, 17, 50, 38, 58, 76, 27, 42, 244, 44, 56, 31, 70, 48, 25, 29, 100, 66, 35, 54, 39, 136, 43, 106, 88, 60, 118, 62, 94, 37, 51, 78, 47, 68, 86

Offset: 0

Jennifer Buckley, Sep 13 2019

nonn

Jennifer Buckley, Table of n, a(n) for n = 0..10000

Inverse: A327119.

a(n) = { my (d=0); while (n, d = select(t -> t>d, divisors(n))[1]; my (k=n/d); if (k%2, n-=d, n+=d)); return (d+1) } \\ Rémy Sigrist, Sep 14 2019

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.

Original entry on oeis.org

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

Jennifer Buckley, Sep 13 2019

nonn

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

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

Cf. A004442, A327093, A327420.

Inverse: A327120.

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
}

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

A327093 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.

Original entry on oeis.org

2, 3, 7, 5, 11, 13, 15, 10, 17, 19, 23, 25, 27, 21, 40, 16, 35, 36, 39, 37, 58, 33, 47, 50, 52, 43, 45, 34, 59, 78, 63, 31, 76, 55, 82, 67, 75, 57, 99, 56, 83, 112, 87, 61, 126, 69, 95, 92, 97, 96, 133, 71, 107, 81, 142, 79, 139, 91, 119, 155, 123, 93, 122, 51, 151, 146, 135

Offset: 1

Jennifer Buckley, Sep 13 2019

nonn

Start with the sequence of positive integers [1, 2, 3, 4, 5, 6, 7, 8, ...].
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).
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).
Swap all pairs specified by k=3, that is, do the swaps (6,3),(12,9),(18,15),(24,21),... .
Continue for all values of k.
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?
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

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

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

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

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
}

def a(n):
    for k in srange(n, 0, -1):
        if k.divides(n):
            n += k if is_odd(n//k) else -k
    return n
print([a(n) for n in (1..67)]) # Peter Luschny, Sep 14 2019

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User: Jennifer Buckley

A372297 Limit of the recursion B(k) = T[k](B(k-1)), where B(1) = (1,2,3,4,5,...) and T[k] is the transformation that permutes the entries k(2i-1) and k(2i) for all positive integers i, if k is prime.

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A327120 Inverse permutation to A327119.

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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.

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A327093 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.

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SageMath

cp's OEIS Frontend

User: Jennifer Buckley

A372297 Limit of the recursion B(k) = T[k](B(k-1)), where B(1) = (1,2,3,4,5,...) and T[k] is the transformation that permutes the entries k(2i-1) and k(2i) for all positive integers i, if k is prime.

Views

Author

Keywords

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Examples

Links

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Programs

Mathematica

A327120 Inverse permutation to A327119.

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Programs

PARI

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.

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Formula

A327093 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.

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SageMath

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.

A327093 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.