A282442 -id:A282442

A282444

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

+20
2

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

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

LINKS

Table of n, a(n) for n=1..19.

Mathematics Stack Exchange user Sheljohn, A curious sequence.

PROG

(Python)

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

CROSSREFS

Cf. A282442.

KEYWORD

nonn,more

AUTHOR

Peter Kagey, Feb 15 2017

EXTENSIONS

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

STATUS

approved

A282434

Positions of records in A282442.

+20
1

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

Equivalently, positions of records in A282443.

LINKS

Peter Kagey, Table of n, a(n) for n = 1..1000

Mathematics Stack Exchange user Sheljohn, A curious sequence.

CROSSREFS

Cf. A282442.

KEYWORD

nonn

AUTHOR

Peter Kagey, Feb 15 2017

STATUS

approved

A282427

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

+20
0

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

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.

LINKS

Table of n, a(n) for n=1..12.

Mathematics Stack Exchange user Sheljohn, A curious sequence.

CROSSREFS

Cf. A282442.

KEYWORD

nonn,more

AUTHOR

Peter Kagey, Feb 15 2017

STATUS

approved

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.

+10
4

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

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

FORMULA

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

EXAMPLE

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.

CROSSREFS

Cf. A282442, A282444.

KEYWORD

nonn

AUTHOR

Peter Kagey, Feb 15 2017

STATUS

approved

A329231

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

+10
4

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,5

COMMENTS

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.

LINKS

Peter Kagey, Table of n, a(n) for n = 1..2048

Mathematics Stack Exchange User Vepir, Grasshopper jumping on circles

CROSSREFS

Cf. A282442, A329230, A329232, A329233.

KEYWORD

nonn,walk

AUTHOR

Peter Kagey, Nov 08 2019

STATUS

approved

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.

+10
1

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

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

EXAMPLE

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.

CROSSREFS

Cf. A282442, A282443.

KEYWORD

nonn

AUTHOR

Peter Kagey, Feb 18 2017

STATUS

approved

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.

+10
1

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,4

COMMENTS

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).

LINKS

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

EXAMPLE

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).

CROSSREFS

Cf. A282442, A282443, A282444.

KEYWORD

nonn

AUTHOR

Peter Kagey, Feb 18 2017

STATUS

approved