A064627 -id:A064627 - cp's OEIS frontend

A064494 Shotgun (or Schrotschuss) numbers: 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.

1, 4, 8, 6, 12, 14, 16, 9, 18, 20, 24, 26, 28, 22, 39, 15, 36, 35, 40, 38, 57, 34, 48, 49, 51, 44, 46, 33, 60, 77, 64, 32, 75, 56, 81, 68, 76, 58, 100, 55, 84, 111, 88, 62, 125, 70, 96, 91, 98, 95, 134, 72, 108, 82, 141, 80, 140, 92, 120, 156, 124, 94, 121, 52, 152, 145

Offset: 1

Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Oct 16 2001

Sequence is prime-free.

			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) = 1,4,8,6,5,3,7,2,10,12,11,14,13,16,...

K. Strassburger, Plot of shotgun numbers

Cf. A064728, A064590, A064627.

max = 66; b[1, j_] := j; b[k_, j_] := b[k, j] = b[k-1, j]; Do[b[k, 2j*k-k] = b[k-1, 2j*k]; b[k, 2j*k] = b[k-1, 2j*k-k], {k, 2, max}, {j, 1, max}]; a[n_] := b[max, n]; Table[a[n], {n, 1, max}] (* Jean-François Alcover, Oct 11 2012 *)

def divsign(s, k):
    if not k.divides(s): return 0
    return (-1)^(s//k)*k
def A064494(n):
    s = n
    for k in srange(n, 1, -1):
        s -= divsign(s, k)
    return s
print([A064494(n) for n in (1..66)]) # Peter Luschny, Sep 16 2019

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

A327446 Numbers missing from A327093.

Original entry on oeis.org

1, 4, 6, 8, 9, 12, 14, 18, 20, 22, 24, 26, 28, 29, 30, 32, 38, 41, 42, 44, 46, 48, 49, 53, 54, 60, 62, 64, 65, 66, 68, 70, 72, 73, 74, 77, 80, 84, 85, 86, 90, 94, 98, 100, 102, 104, 106, 108, 109, 110, 111, 114, 116, 118, 120, 124, 125, 128, 130, 132, 136, 137, 138, 140, 149, 150

Offset: 1

N. J. A. Sloane, Sep 14 2019

nonn

It would be nice to have an alternative characterization of these numbers.

Cf. A327093, A327445, A327419, A064494, A064627, A327420.

# Use with caution: search range must be adjusted as necessary!
def A327446List(size):
    return sorted(Set([A327419(n) for n in (1..3*size)]))[0:size]
print(A327446List(66)) # Peter Luschny, Sep 16 2019

A266679 Positive integers not shotgun (or Schrotschuss) numbers, in order of the first number to be permuted forward by the transformations T[k] where k = 2 or k is odd.

Original entry on oeis.org

2, 3, 5, 7, 10, 11, 13, 21, 17, 19, 30, 23, 27, 25, 29, 31, 45, 42, 37, 54, 41, 43, 65, 47, 50, 69, 53, 66, 78, 59, 61, 63, 86, 67, 93, 71, 73, 105, 85, 79, 74, 83, 110, 117, 89, 112, 126, 115, 97, 99, 101

Offset: 1

Timothy Mott, Jan 02 2016

nonn

In the construction of the shotgun numbers (A064494), the first element to be shifted forward by T[k] where k = 2 or k is odd will also be shifted forward by subsequent transformations T[2^n*k], and will therefore not appear in that sequence. In particular, no prime numbers appear in A064494, and an odd prime p is element (p+1)/2 in this list.

Cf. A064494. Sorted gives A064627.

cp's OEIS Frontend

A064494 Shotgun (or Schrotschuss) numbers: 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.

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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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A327446 Numbers missing from A327093.

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A266679 Positive integers not shotgun (or Schrotschuss) numbers, in order of the first number to be permuted forward by the transformations T[k] where k = 2 or k is odd.

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cp's OEIS Frontend

A064494 Shotgun (or Schrotschuss) numbers: 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.

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Programs

Mathematica

SageMath

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

A327446 Numbers missing from A327093.

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Author

Keywords

Comments

Crossrefs

Programs

SageMath

A266679 Positive integers not shotgun (or Schrotschuss) numbers, in order of the first number to be permuted forward by the transformations T[k] where k = 2 or k is odd.

Views

Author

Keywords

Comments

Crossrefs

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.