A282442 -id:A282442 - cp's OEIS frontend

A282444 Numbers k such that A282442(k) = k + 1.

1, 2, 5, 8, 14, 50, 119, 200, 269, 299, 1154, 5369, 47249, 48299, 58643, 130325, 148979, 282074, 887480

Offset: 1

Peter Kagey, Feb 15 2017

a(18) > 10^6. - Hakan Icoz, Apr 09 2021

Mathematics Stack Exchange user Sheljohn, A curious sequence.

def A282442():
    n = 0
    while True:
        n += 1
        h = n
        k = 2
        while h >= k:
            h = h % k
            h = n - h
            k += 1
        yield k
n=0
for i in A282442():
    n += 1
    if i == n+1:
        print(n) # Hakan Icoz, Apr 09 2021

a(13)-a(19) from Hakan Icoz, Apr 09 2021

A282434 Positions of records in A282442.

Original entry on oeis.org

1, 2, 4, 5, 8, 11, 12, 14, 18, 19, 21, 24, 25, 28, 34, 40, 41, 50, 54, 55, 60, 63, 70, 76, 86, 87, 90, 96, 99, 107, 118, 119, 132, 139, 152, 164, 181, 184, 190, 197, 200, 208, 220, 233, 236, 237, 242, 252, 269, 272, 285, 288, 298, 299, 324, 328, 341, 354, 357

Offset: 1

Peter Kagey, Feb 15 2017

nonn

Equivalently, positions of records in A282443.

Peter Kagey, Table of n, a(n) for n = 1..1000
Mathematics Stack Exchange user Sheljohn, A curious sequence.

Cf. A282442.

A282427 Numbers k such that A282442(k) = ceiling(k/2) + 1.

Original entry on oeis.org

1, 3, 7, 39, 47, 111, 959, 3319, 7407, 11967, 13007, 16239

Offset: 1

Peter Kagey, Feb 15 2017

All terms are odd.

Proof: if A282442(2m) = m + 1, then the step of length m would have to have concluded on exactly the middle step, but a phase with step-length m cannot end on the middle step because the distance from the middle step to the top/bottom of the staircase is equal to m.

Mathematics Stack Exchange user Sheljohn, A curious sequence.

Cf. A282442.

A282443 a(n) is the largest step size that is taken on a staircase of n steps when following the following procedure: Take steps of length 1 up a staircase until you can't step any further, then take steps of length 2 down until you can't step any further, and so on.

Original entry on oeis.org

1, 2, 2, 3, 5, 4, 4, 8, 8, 7, 9, 10, 10, 14, 14, 10, 11, 17, 18, 15, 19, 16, 14, 23, 24, 17, 19, 27, 18, 23, 25, 20, 20, 30, 30, 19, 27, 24, 20, 31, 39, 32, 30, 38, 38, 24, 24, 34, 34, 50, 46, 31, 39, 53, 54, 47, 49, 40, 38, 59, 58, 57, 62, 58, 48, 49, 57, 39

Offset: 1

Peter Kagey, Feb 15 2017

nonn

			For n = 4:
step size 1: 0 -> 1 -> 2 -> 3 -> 4;
step size 2: 4 -> 2 -> 0;
step size 3: 0 -> 3.
Because the walker cannot take four steps down, the biggest step size is 3.
Therefore a(4) = 3.

Peter Kagey, Table of n, a(n) for n = 1..10000

Cf. A282442, A282444.

a(n) = A282442(n) - 1.

A329231 The maximum number of times one reaches a single position during the grasshopper procedure.

Original entry on oeis.org

1, 1, 1, 1, 3, 2, 1, 1, 4, 3, 3, 3, 3, 3, 3, 1, 5, 2, 3, 3, 4, 3, 5, 3, 3, 4, 5, 3, 4, 4, 4, 1, 4, 4, 3, 4, 4, 3, 3, 5, 4, 5, 3, 3, 4, 4, 5, 4, 6, 4, 5, 4, 5, 4, 5, 4, 4, 4, 5, 5, 4, 5, 5, 1, 4, 4, 5, 3, 4, 5, 5, 4, 4, 7, 4, 4, 4, 5, 5, 5, 4, 4, 4, 4, 4, 5, 4

Offset: 1

Peter Kagey, Nov 08 2019

The grasshopper procedure: n positions are evenly spaced around a circle, a grasshopper hops randomly to any position, after the k-th hop, the grasshopper looks clockwise and counterclockwise k positions. If one of the positions has been visited less often then the other, it hops there; if both positions have been visited an equal number of times, it hops k steps in the clockwise position. (See Mathematics Stack Exchange link for more details.)
a(n) >= (A329230(n)-1)/(n-1).
Least values of n such that a(n) = 1, 2, 3, etc are 1, 6, 5, 9, 17, 49, 74, 198, 688, 1745 etc.
Conjecture: a(n) = 1 if and only if n = 3, n = 7, or n = 2^k for some k.
Conjecture: The largest values of n for which a(n) = 2, 3, 4, 5 respectively are n = 18, 68, 381, 1972.
If the second conjecture is true, then 2, 3, 4, and 5 appear 2, 19, 87, and 313 times respectively.
Conjecture: Every integer greater than 1 appears in this sequence a finite number of times.

Peter Kagey, Table of n, a(n) for n = 1..2048
Mathematics Stack Exchange User Vepir, Grasshopper jumping on circles

Cf. A282442, A329230, A329232, A329233.

A282573 The number of steps taken on a staircase of n steps during the following routine: Take steps of length 1 up a staircase until you can't step any further, then take steps of length 2 down until you can't step any further, and so on.

Original entry on oeis.org

1, 3, 4, 7, 10, 12, 13, 19, 20, 23, 26, 32, 33, 39, 40, 41, 46, 53, 57, 56, 63, 65, 66, 77, 81, 80, 83, 94, 90, 97, 100, 102, 103, 117, 118, 117, 128, 126, 127, 138, 149, 151, 152, 162, 163, 160, 161, 175, 176, 194, 195, 186, 197, 212, 216, 215, 218, 220, 221

Offset: 1

Peter Kagey, Feb 18 2017

nonn

			For n = 4:
step size 1: 0 -> 1 -> 2 -> 3 -> 4 (four steps);
step size 2: 4 -> 2 -> 0 (two steps);
step size 3: 0 -> 3 (one step).
Because the walker cannot take four steps down, a(4) = 4 + 2 + 1 = 7.

Peter Kagey, Table of n, a(n) for n = 1..10000

Cf. A282442, A282443.

A282574 The final position on a staircase of n steps during the following routine: Take steps of length 1 up a staircase until you can't step any further, then take steps of length 2 down until you can't step any further, and so on.

Original entry on oeis.org

1, 0, 1, 3, 5, 2, 3, 0, 1, 7, 9, 2, 3, 0, 1, 6, 11, 17, 1, 15, 19, 6, 9, 23, 1, 17, 19, 27, 11, 23, 25, 12, 13, 4, 5, 19, 27, 14, 19, 31, 39, 10, 13, 6, 7, 22, 23, 14, 15, 0, 5, 31, 39, 53, 1, 47, 49, 18, 21, 59, 3, 57, 1, 6, 17, 49, 57, 39, 43, 69, 9, 47, 51

Offset: 1

Peter Kagey, Feb 18 2017

nonn

If a(n) = 0 or a(n) = n, then A282443(n) = n and n is in A282444.

a(n) is bounded above by A282443(n) and bounded below by n - A282443(n).

			For n = 4:
step size 1: 0 -> 1 -> 2 -> 3 -> 4 (four steps);
step size 2: 4 -> 2 -> 0 (two steps);
step size 3: 0 -> 3 (one step).
Because the walker cannot take four steps down, a(4) = 3 (the final position).

Peter Kagey, Table of n, a(n) for n = 1..10000

Cf. A282442, A282443, A282444.

cp's OEIS Frontend

A282444 Numbers k such that A282442(k) = k + 1.

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A282434 Positions of records in A282442.

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A282427 Numbers k such that A282442(k) = ceiling(k/2) + 1.

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A282443 a(n) is the largest step size that is taken on a staircase of n steps when following the following procedure: Take steps of length 1 up a staircase until you can't step any further, then take steps of length 2 down until you can't step any further, and so on.

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A329231 The maximum number of times one reaches a single position during the grasshopper procedure.

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A282573 The number of steps taken on a staircase of n steps during the following routine: Take steps of length 1 up a staircase until you can't step any further, then take steps of length 2 down until you can't step any further, and so on.

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Author

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A282574 The final position on a staircase of n steps during the following routine: Take steps of length 1 up a staircase until you can't step any further, then take steps of length 2 down until you can't step any further, and so on.

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