A087706 -id:A087706 - cp's OEIS frontend

A087704 Number of steps for iteration of map x -> (5/3)*floor(x) to reach an integer > n when started at n, or -1 if no such integer is ever reached.

2, 1, 2, 4, 1, 3, 3, 1, 9, 2, 1, 2, 4, 1, 8, 5, 1, 3, 2, 1, 2, 3, 1, 9, 7, 1, 4, 2, 1, 2, 5, 1, 3, 3, 1, 4, 2, 1, 2, 8, 1, 6, 4, 1, 3, 2, 1, 2, 3, 1, 5, 4, 1, 6, 2, 1, 2, 7, 1, 3, 3, 1, 6, 2, 1, 2, 7, 1, 4, 5, 1, 3, 2, 1, 2, 3, 1, 4, 7, 1, 10, 2, 1, 2, 4, 1, 3, 3, 1, 5, 2, 1, 2, 4, 1, 8, 6, 1, 3

Offset: 2

N. J. A. Sloane, Sep 29 2003

nonn

It is conjectured that an integer is always reached.

Robert Israel, Table of n, a(n) for n = 2..10000
J. C. Lagarias and N. J. A. Sloane, Approximate squaring (pdf, ps), Experimental Math., 13 (2004), 113-128.

Cf. A087705, A087706, A087707.

f2 := proc(x,y) x*floor(y); end; r := 5/3; h := proc(x) local n,y; global r; y := f2(r,x); for n from 1 to 20 do if whattype(y) = 'integer' then RETURN([x,n,y]); else y := f2(r,y); fi; od: RETURN(['NULL','NULL','NULL']); end; [seq(h(n)[2],n=2..60)];

from fractions import Fraction
def A087704(n):
    x, c = Fraction(n,1), 0
    while x.denominator > 1 or x<=n:
        x = Fraction(5*x._floor_(),3)
        c += 1
    return c # Chai Wah Wu, Sep 01 2023

a(n) = a(n + m) if a(n) > 0 and m is a (positive or negative) multiple of 3^a(n). - Robert Israel, Sep 01 2023

A087705 First integer > n reached under iteration of map x -> (5/3)*floor(x) when started at n, or -1 if no such integer is ever reached.

Original entry on oeis.org

5, 5, 10, 35, 10, 30, 35, 15, 905, 30, 20, 35, 105, 25, 905, 210, 30, 85, 55, 35, 60, 105, 40, 2410, 905, 45, 210, 80, 50, 85, 405, 55, 155, 160, 60, 280, 105, 65, 110, 2410, 70, 905, 335, 75, 210, 130, 80, 135, 230, 85, 660, 405, 90, 1160, 155, 95, 160, 2085, 100

Offset: 2

N. J. A. Sloane, Sep 29 2003

nonn

It is conjectured that an integer is always reached.

Robert Israel, Table of n, a(n) for n = 2..10000
J. C. Lagarias and N. J. A. Sloane, Approximate squaring (pdf, ps), Experimental Math., 13 (2004), 113-128.

Cf. A087704, A087706.

f2 := proc(x,y) x*floor(y); end; r := 5/3; h := proc(x) local n,y; global r; y := f2(r,x); for n from 1 to 20 do if whattype(y) = 'integer' then RETURN([x,n,y]); else y := f2(r,y); fi; od: RETURN(['NULL','NULL','NULL']); end; [seq(h(n)[3],n=2..60)];

from fractions import Fraction
def A087705(n):
    x = Fraction(n,1)
    while x.denominator > 1 or x<=n:
        x = Fraction(5*x._floor_(),3)
    return int(x) # Chai Wah Wu, Sep 01 2023

A087707 Number of steps for iteration of map x -> (5/3)*ceiling(x) to reach an integer > n when started at n, or -1 if no such integer is ever reached.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Sep 29 2003

nonn

It is conjectured that an integer is always reached.

J. C. Lagarias and N. J. A. Sloane, Approximate squaring (pdf, ps), Experimental Math., 13 (2004), 113-128.

Cf. A087704, A087705, A087706, A087708, A087709.

c2 := proc(x,y) x*ceil(y); end; r := 5/3; ch := proc(x) local n,y; global r; y := c2(r,x); for n from 1 to 20 do if whattype(y) = 'integer' then RETURN([x,n,y]); else y := c2(r,y); fi; od: RETURN(['NULL','NULL','NULL']); end; [seq(ch(n)[2],n=1..100)];

from fractions import Fraction
def A087707(n):
    x, c = Fraction(n), 0
    while x.denominator > 1 or x<=n:
        x = Fraction(5*x._ceil_(),3)
        c += 1
    return c # Chai Wah Wu, Sep 02 2023

A087708 First integer > n reached under iteration of map x -> (5/3)*ceiling(x) when started at n, or -1 if no such integer is ever reached.

Original entry on oeis.org

20, 20, 5, 20, 15, 10, 20, 40, 15, 1770, 90, 20, 290, 40, 25, 45, 1770, 30, 90, 95, 35, 290, 65, 40, 70, 345, 45, 220, 1770, 50, 145, 90, 55, 95, 165, 60, 290, 17845, 65, 520, 115, 70, 120, 345, 75, 215, 220, 80, 1770, 140, 85, 145, 415, 90, 715, 1215, 95, 270, 165, 100, 170

Offset: 1

N. J. A. Sloane, Sep 29 2003

nonn

It is conjectured that an integer is always reached.

J. C. Lagarias and N. J. A. Sloane, Approximate squaring (pdf, ps), Experimental Math., 13 (2004), 113-128.

Cf. A087704, A087705, A087706, A087707, A087709.

c2 := proc(x,y) x*ceil(y); end; r := 5/3; ch := proc(x) local n,y; global r; y := c2(r,x); for n from 1 to 20 do if whattype(y) = 'integer' then RETURN([x,n,y]); else y := c2(r,y); fi; od: RETURN(['NULL','NULL','NULL']); end; [seq(ch(n)[3],n=1..100)];

from fractions import Fraction
def A087708(n):
    x = Fraction(n)
    while x.denominator > 1 or x<=n:
        x = Fraction(5*x._ceil_(),3)
    return int(x) # Chai Wah Wu, Sep 02 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

A365368 First integer > n reached under iteration of map x -> (5/3)*round(x) when started at n, or -1 if no such integer is ever reached.

Original entry on oeis.org

5, 5, 5, 20, 10245, 10, 20, 10245, 15, 130, 30, 20, 10245, 105, 25, 45, 130, 30, 245, 55, 35, 10245, 105, 40, 70, 120, 45, 130, 80, 50, 145, 245, 55, 95, 270, 60, 10245, 105, 65, 520, 870, 70, 120, 2605, 75, 355, 130, 80, 380, 230, 85, 145, 245, 90, 255, 155, 95

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, A365341, A365342, A365367.

from fractions import Fraction
def A365368(n):
    x = Fraction(n)
    while x.denominator > 1 or x<=n:
        x = Fraction(5*x._round_(),3)
    return int(x)

A365369 A365368/5, except when A365368(n) = -1, then a(n) = -1.

Original entry on oeis.org

1, 1, 1, 4, 2049, 2, 4, 2049, 3, 26, 6, 4, 2049, 21, 5, 9, 26, 6, 49, 11, 7, 2049, 21, 8, 14, 24, 9, 26, 16, 10, 29, 49, 11, 19, 54, 12, 2049, 21, 13, 104, 174, 14, 24, 521, 15, 71, 26, 16, 76, 46, 17, 29, 49, 18, 51, 31, 19, 54, 151, 20, 34, 2049, 21, 99, 36

Offset: 1

Chai Wah Wu, Sep 02 2023

nonn

Cf. A087704, A087705, A087706, A087707, A087708, A087709, A365341, A365342, A365367, A365368.

from fractions import Fraction
def A365369(n):
    x = Fraction(n)
    while x.denominator > 1 or x<=n:
        x = Fraction(5*x._round_(),3)
    return int(x)//5

A365370 Positions of records in A365367.

Original entry on oeis.org

1, 5, 415, 635, 15935, 60971, 275039, 514661, 2857994, 14179544, 170794880, 2382918520

Offset: 1

Chai Wah Wu, Sep 02 2023

nonn

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

Cf. A087704, A087705, A087706, A087707, A087708, A087709, A365341, A365342, A365367, A365368.

A365367(a(n)) = A365371(n).

A365371 Record values of A365367.

Original entry on oeis.org

3, 15, 17, 21, 24, 28, 32, 35, 37, 45, 50, 55

Offset: 1

Chai Wah Wu, Sep 02 2023

nonn

Numbers v = A365367(k) such that A365367(m) < v for 1 <= m < k.

Cf. A087704, A087705, A087706, A087707, A087708, A087709, A365341, A365342, A365367, A365368, A365369, A365370.

a(n) = A365367(A365370(n)).

cp's OEIS Frontend

A087704 Number of steps for iteration of map x -> (5/3)*floor(x) to reach an integer > n when started at n, or -1 if no such integer is ever reached.

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A087705 First integer > n reached under iteration of map x -> (5/3)*floor(x) when started at n, or -1 if no such integer is ever reached.

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A087707 Number of steps for iteration of map x -> (5/3)*ceiling(x) to reach an integer > n when started at n, or -1 if no such integer is ever reached.

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A087708 First integer > n reached under iteration of map x -> (5/3)*ceiling(x) when started at n, or -1 if no such integer is ever reached.

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Maple

Python

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.

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A365368 First integer > n reached under iteration of map x -> (5/3)*round(x) when started at n, or -1 if no such integer is ever reached.

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A365369 A365368/5, except when A365368(n) = -1, then a(n) = -1.

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A365370 Positions of records in A365367.

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A365371 Record values of A365367.

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