A327420 -id:A327420 - cp's OEIS frontend

A327487 T(n, k) are the summands given by the generating function of A327420(n), triangle read by rows, T(n,k) for 0 <= k <= n.

1, 2, -2, 3, -3, 2, 4, -4, 3, 0, 5, -5, 4, 0, 2, 6, -6, 5, 0, 0, 0, 7, -7, 6, 0, 0, 3, 0, 8, -8, 7, 0, 0, 0, 0, 0, 9, -9, 8, 0, 0, 0, 4, 3, 0, 10, -10, 9, 0, 0, 0, 0, 0, -3, -2, 11, -11, 10, 0, 0, 0, 0, 5, 0, -3, 2, 12, -12, 11, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Peter Luschny, Sep 14 2019

			Triangle starts (at the end of the line is the row sum (A327420)):
[ 0] [ 1] 1
[ 1] [ 2,  -2] 0
[ 2] [ 3,  -3,  2] 2
[ 3] [ 4,  -4,  3, 0] 3
[ 4] [ 5,  -5,  4, 0, 2] 6
[ 5] [ 6,  -6,  5, 0, 0, 0] 5
[ 6] [ 7,  -7,  6, 0, 0, 3, 0] 9
[ 7] [ 8,  -8,  7, 0, 0, 0, 0, 0] 7
[ 8] [ 9,  -9,  8, 0, 0, 0, 4, 3,  0] 15
[ 9] [10, -10,  9, 0, 0, 0, 0, 0, -3, -2] 4
[10] [11, -11, 10, 0, 0, 0, 0, 5,  0, -3, 2] 14

Cf. A327420, A327093, A057032, A069829.

def divsign(s, k):
    if not k.divides(s): return 0
    return (-1)^(s//k)*k
def A327487row(n):
    s = n + 1
    r = srange(s, 1, -1)
    S = [-divsign(s, s)]
    for k in r:
        s += divsign(s, k)
        S.append(-divsign(s, k))
    return S
# Prints the triangle like in the example section.
for n in (0..10):
    print([n], A327487row(n), sum(A327487row(n)))

Sum_{k=0..n} T(n, k) = A327420(n).

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

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

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

cp's OEIS Frontend

A327487 T(n, k) are the summands given by the generating function of A327420(n), triangle read by rows, T(n,k) for 0 <= k <= n.

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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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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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A327446 Numbers missing from A327093.

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cp's OEIS Frontend

A327487 T(n, k) are the summands given by the generating function of A327420(n), triangle read by rows, T(n,k) for 0 <= k <= n.

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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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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

A327446 Numbers missing from A327093.

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Author

Keywords

Comments

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Programs

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.

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.