A365343 -id:A365343 - cp's OEIS frontend

A365342 Positions of records in A087704.

2, 5, 10, 82, 284, 680, 1322, 68104, 149795, 213895, 1023127, 3775307, 25396927, 36254395, 53343289, 68677522, 266888359, 366901277, 558829814, 1576699732, 8527370677, 11616255230, 16948492520, 167299409017, 222801579737, 2001199132825, 5024272986979, 7880897129684

Offset: 1

Robert Israel, Sep 01 2023

nonn

Numbers k such that iteration of the map x -> (5/3)*floor(x) starting at x = k takes more steps to reach an integer > k than it does for any number from 2 to k - 1.

			a(3) = 10 is a term because A087704(10) = 9 and A087704(k) < 9 for 2 <= k < 10.

Cf. A087704, A365343.

g:= x -> 5/3 * floor(x):
h:= proc(n) local i,k;
  k:= g(n);
  for i from 1 while not (k::integer and k > n) do k:= g(k) od:
  i
end proc:
M:= 2: A:= 2: count:= 1:
for n from 3 while count < 17  do
  v:= h(n);
  if v > M then count:= count+1; A:= A,n; M:= v fi;
od:
A;

g =  5/3 * Floor[#]&;
h[n_] := Module[{i, k}, k = g[n]; For[i = 1, !IntegerQ[k] && k > n, i++,  k = g[k]]; i];
M = 2; A = {2}; count = 1;
For[n = 3, count < 17, n++, v = h[n]; If[v > M, count++; A = Append[A, n]; Print[A]; M = v]];
A (* Jean-François Alcover, Sep 14 2023, after Robert Israel *)

A087704(a(n)) = A365343(n).

a(18)-a(21) from Chai Wah Wu, Sep 02 2023

a(22)-a(28) from Martin Ehrenstein, Sep 03 2023

A365367 Number of steps for iteration of map x -> (5/3)*round(x) to reach an integer > n when started at n, or -1 if no such integer is ever reached.

Original entry on oeis.org

3, 2, 1, 3, 15, 1, 2, 14, 1, 5, 2, 1, 13, 4, 1, 2, 4, 1, 5, 2, 1, 12, 3, 1, 2, 3, 1, 3, 2, 1, 3, 4, 1, 2, 4, 1, 11, 2, 1, 5, 6, 1, 2, 8, 1, 4, 2, 1, 4, 3, 1, 2, 3, 1, 3, 2, 1, 3, 5, 1, 2, 10, 1, 4, 2, 1, 4, 5, 1, 2, 6, 1, 7, 2, 1, 5, 3, 1, 2, 3, 1, 3, 2, 1, 3

Offset: 1

Chai Wah Wu, Sep 02 2023

nonn

Conjecture: an integer will always be reached, i.e. a(n) > 0 for all n.

Cf. A087704, A087705, A087706, A087707, A087708, A087709, A365342, A365343.

from fractions import Fraction
def A365367(n):
    x, c = Fraction(n), 0
    while x.denominator > 1 or x<=n:
        x = Fraction(5*x._round_(),3)
        c += 1
    return c

A365344 a(0) = 0; for n > 0, a(n) is the largest distance squared on a square spiral between a(n-1) and any previous occurrence of a(n-1). If a(n-1) has not previously occurred then a(n) = 0.

Original entry on oeis.org

0, 0, 1, 0, 2, 0, 4, 0, 4, 4, 9, 0, 10, 0, 10, 4, 18, 0, 16, 0, 20, 0, 20, 4, 20, 16, 29, 0, 26, 0, 34, 0, 34, 4, 26, 20, 41, 0, 40, 0, 40, 4, 34, 72, 0, 45, 0, 41, 61, 0, 74, 0, 58, 0, 50, 0, 61, 50, 5, 0, 58, 32, 0, 85, 0, 113, 0, 89, 0, 73, 0, 89, 16, 53, 0, 85, 68, 0, 89, 61, 65, 0, 145, 0

Offset: 0

Scott R. Shannon, Oct 16 2023

nonn

			The spiral begins:
.
  41--20--26--4---34--0---34  .
  |                       |   .
  0   18--4---10--0---10  0   50
  |   |               |   |   |
  40  0   2---0---1   0   26  0
  |   |   |       |   |   |   |
  0   16  0   0---0   9   0   58
  |   |   |           |   |   |
  40  0   4---0---4---4   29  0
  |   |                   |   |
  4   20--0---20--4---20--16  74
  |                           |
  34--72--0---45--0---41--61--0
.
.
a(4) = 2 as a(3) = 0 and the largest square distance between a(3) and a previous occurrence of 0 is 2 - between a(3) and a(1).
a(47) = 41 as a(46) = 0 and the largest square distance between a(46) and a previous occurrence of 0 is 41 - between a(46) and a(37). This is the first term to differ from A365343.

Scott R. Shannon, Table of n, a(n) for n = 0..10000

Cf. A366543, A366353, A366354, A001481.

cp's OEIS Frontend

A365342 Positions of records in A087704.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A365367 Number of steps for iteration of map x -> (5/3)*round(x) to reach an integer > n when started at n, or -1 if no such integer is ever reached.

Views

Author

Keywords

Comments

Crossrefs

Programs

Python

A365344 a(0) = 0; for n > 0, a(n) is the largest distance squared on a square spiral between a(n-1) and any previous occurrence of a(n-1). If a(n-1) has not previously occurred then a(n) = 0.

Views

Author

Keywords

Examples

Links

Crossrefs