A325606 -id:A325606 - cp's OEIS frontend

A157428 Lower left-hand x-coordinates of invisible 2 X 2 squares with 0 < y < x.

54, 174, 550, 574, 588, 608, 678, 740, 740, 790, 798, 804, 874, 944, 986, 986, 994, 1034, 1064, 1104, 1104, 1158, 1208, 1274, 1308, 1308, 1308, 1308, 1406, 1420, 1462, 1462, 1462, 1494, 1494, 1550, 1580, 1580, 1610, 1624, 1638, 1650, 1664, 1664, 1664

Offset: 1

Eric W. Weisstein, Mar 01 2009

nonn

			{54, 20}, {174, 98}, {550, 114}, {574, 368}, {588, 494}, {608, 474}, ...

Eric Weisstein's World of Mathematics, Visible Point

Cf. A157426, A157427, A157429, A325602, A325603, A325604, A325605, A325606, A325607.

A157426 Lower left-hand x-coordinates of invisible 1x1 squares with 0 < y < x.

Original entry on oeis.org

20, 35, 35, 54, 65, 69, 77, 84, 84, 95, 98, 98, 99, 99, 104, 104, 104, 110, 114, 114, 119, 119, 132, 132, 135, 135, 147, 153, 153, 159, 161, 170, 170, 170, 174, 174, 175, 185, 185, 186, 186, 188, 189, 189, 189, 189, 189, 195, 195, 195

Offset: 1

Eric W. Weisstein, Mar 01 2009

nonn

			{20, 14}, {35, 14}, {35, 20}, {54, 44}, {65, 39}, {69, 45}, {77, 21}, {84, 34}, {84, 50}, {95, 75}, ...

Eric Weisstein's World of Mathematics, Visible Point

Cf. A157426, A157428, A157429, A325602.

Cf. A325603, A325604, A325605, A325606, A325607.

A325604 Lower left-hand x-coordinate for 3 X 3 invisible forest with 0 < x < y.

Original entry on oeis.org

1274, 1884, 1924, 1448, 594, 2254, 2364, 2210, 1598, 1000, 2624, 740, 664, 2408, 1924, 1924, 494, 1000, 230, 2914, 650, 1000, 644, 3794, 1924, 3794, 2430, 104, 4146, 3534, 4234, 1748, 2254, 2408, 1274, 1064, 4640, 4520, 2484, 3794

Offset: 1

Benjamin Hutz, May 10 2019

nonn

These are 3 X 3 rectangles of lattice points not visible along straight lines of sight from the origin. The sequence is ordered by Euclidean distance from (0,0).

			(1274,1308), (1884,2000), (1924,2330), (1448,2714), (594,3128), (2254,2540), (2364,2924), (2210,3080), (1598,3484), (1000,3794)

Benjamin Hutz, Table of n, a(n) for n = 1..1000
E. Goins, P. Harris, B. Kubik, A. Mbirika, Lattice Point Visibility on Generalized Lines of Sight, arXiv:1710.04554 [math.NT], 2017; Amer. Math. Monthly 125 (2018) 593-601.
F. Herzog, B. M. Stewart, Patterns of Visible and Nonvisible Lattice Points, Amer. Math. Monthly 78 (1971) 487-496
S. Laishram, F. Luca, Rectangles Of Nonvisible Lattice Points, J. Int. Seq. 18 (2015) 15.10.8.

Cf. A157426, A157427, A157428, A157429.

Cf. A325602, A325603, A325605, A325606, A325607.

A331400 The grid points visible from the central point of an infinite 2D square lattice where all grid points are numbered as in the Ulam spiral.

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 16, 18, 20, 22, 24, 26, 27, 29, 30, 32, 33, 35, 36, 38, 39, 41, 42, 44, 45, 47, 48, 50, 52, 54, 56, 58, 60, 62, 64, 66, 68, 70, 72, 74, 76, 78, 80, 82, 83, 84, 85, 87, 88, 89, 90, 92, 93, 94, 95, 97, 98, 99, 100, 102, 103, 104, 105, 107, 108, 109, 110, 112, 113, 114, 115

Offset: 1

Scott R. Shannon, Jan 16 2020

nonn

Any grid point with relative coordinates (x,y) from the central grid point, which is numbered 1, and where the greatest common divisor (gcd) of |x| and |y| equals 1 will be visible from the central point. Grid points where gcd(|x|,|y|) > 1 will have another point directly between it and the central point and will thus not be visible.

In an infinite 2D square lattice the ratio of visible grid points to all points is 6/pi^2, the same as the probability of two random numbers being relative prime.

			a(1) = 2 to a(8) = 9 are the eight adjacent grid points to point 1, thus all are visible from that point.
a(9) = 10 is the first non-adjacent point to square 1, but as it is located at relative coordinates (2,-1) it is visible as gcd(|-2|,|1|) = 1.
The point numbered 11 is the first point not visible from point 1 as it has relative coordinates (2,0) and gcd(|2|,|0|) = 2.

Scott R. Shannon, Table of n, a(n) for n = 1..20000.
Scott R. Shannon, Image showing the grid points visible from point 1 for the first 100000 grid points. Each point is surrounded by a 1 X 1 square which is colored according to the visibility of the point is contains - yellow indicates those points visible from point 1, gray are those not visible. All squares are labeled with their corresponding point number in the Ulam spiral numbering. The spiral path tracing out the number ordering is also shown.
Eric Weisstein's World of Mathematics, Visible Point.
Wikipedia, Ulam Spiral.

Cf. A063826, A157426, A157428, A325604, A325606.

A325602 Lower left-hand x-coordinate for 2 X 2 invisible forest with 0 < x < y.

Original entry on oeis.org

14, 14, 20, 44, 39, 21, 45, 34, 50, 21, 44, 39, 54, 75, 45, 65, 34, 77, 74, 69, 90, 56, 50, 84, 76, 33, 84, 14, 20, 69, 55, 111, 75, 33, 14, 105, 35, 119, 95, 20, 56, 35, 74, 90, 110, 104, 76, 62, 20, 35

Offset: 1

Benjamin Hutz, May 10 2019

nonn

These are 2 X 2 rectangles of lattice points not visible along straight lines of sight from the origin. The sequence is ordered by Euclidean distance from (0,0).

			(14,20), (14,35), (20,35), (44,54), (39,65), (21,77), (45,69), (34,84), ...

Benjamin Hutz, Table of n, a(n) for n = 1..1000
E. Goins, P. Harris, B. Kubik, A. Mbirika, Lattice Point Visibility on Generalized Lines of Sight, arXiv:1710.04554 [math.NT], 2017; Amer. Math. Monthly 125 (2018) 593-601.
F. Herzog, B. M. Stewart, Patterns of Visible and Nonvisible Lattice Points, Amer. Math. Monthly 78 (1971) 487-496
S. Laishram, F. Luca, Rectangles Of Nonvisible Lattice Points, J. Int. Seq. 18 (2015) 15.10.8.

Cf. A157426, A157427, A157428, A157429.

Cf. A325603, A325604, A325605, A325606, A325607.

def is_nxn(x,y,n):
    if all([gcd(x+a,y+b) != 1 for a in range(n) for b in range(n)]):
        return True
    return False
def insert_item(pts, item, index):
    N = len(pts)
    if N == 0:
      return [item]
    elif N == 1:
        if item[index] < pts[0][index]:
            pts.insert(0,item)
        else:
            pts.append(item)
        return pts
    else: #binary insertion
        left = 1
        right = N
        mid = ((left + right)/2).floor()
        if item[index] < pts[mid][index]:
        # item goes into first half
            return insert_item(pts[:mid], item, index) + pts[mid:N]
        else:
        # item goes into second half
            return pts[:mid] + insert_item(pts[mid:N], item, index)
B=1200
L=[]
for x in range(1,B):
    for y in range(x+1,B):
        if is_nxn(x,y,n=2):
            G=[x,y,x^2+y^2]
            L=insert_item(L, G, 2)

A325603 Lower left-hand y-coordinate for 2 X 2 invisible forest with 0 < x < y.

Original entry on oeis.org

20, 35, 35, 54, 65, 77, 69, 84, 84, 98, 99, 104, 99, 95, 114, 104, 119, 98, 110, 114, 104, 132, 135, 119, 132, 153, 135, 174, 174, 161, 175, 147, 170, 186, 189, 159, 189, 153, 170, 195, 189, 195, 185, 195, 185, 195, 209, 216, 224, 224

Offset: 1

Benjamin Hutz, May 10 2019

nonn

These are 2 X 2 rectangles of lattice points not visible along straight lines of sight from the origin. The sequence is ordered by Euclidean distance from (0,0).

			(14,20), (14,35), (20,35), (44,54), (39,65), (21,77), (45,69), (34,84).

Benjamin Hutz, Table of n, a(n) for n = 1..1000
E. Goins, P. Harris, B. Kubik, A. Mbirika, Lattice Point Visibility on Generalized Lines of Sight, arXiv:1710.04554 [math.NT], 2017; Amer. Math. Monthly 125 (2018) 593-601.
F. Herzog, B. M. Stewart, Patterns of Visible and Nonvisible Lattice Points, Amer. Math. Monthly 78 (1971) 487-496
S. Laishram, F. Luca, Rectangles Of Nonvisible Lattice Points, J. Int. Seq. 18 (2015) 15.10.8.

Cf. A157426, A157427, A157428, A157429, A325602, A325604, A325605, A325606, A325607.

def is_nxn(x,y,n):
    if all([gcd(x+a,y+b) != 1 for a in range(n) for b in range(n)]):
        return True
    return False
def insert_item(pts, item, index):
    N = len(pts)
    if N == 0:
      return [item]
    elif N == 1:
        if item[index] < pts[0][index]:
            pts.insert(0,item)
        else:
            pts.append(item)
        return pts
    else: #binary insertion
        left = 1
        right = N
        mid = ((left + right)/2).floor()
        if item[index] < pts[mid][index]:
        # item goes into first half
            return insert_item(pts[:mid], item, index) + pts[mid:N]
        else:
        # item goes into second half
            return pts[:mid] + insert_item(pts[mid:N], item, index)
B=1200
L=[]
for x in range(1,B):
    for y in range(x+1,B):
        if is_nxn(x,y,n=2):
            G=[x,y,x^2+y^2]
            L=insert_item(L, G, 2)

A157427 Lower left-hand y-coordinates of invisible 1x1 squares with 0 < y < x.

Original entry on oeis.org

14, 14, 20, 44, 39, 45, 21, 34, 50, 75, 21, 77, 44, 54, 39, 65, 90, 74, 45, 69, 34, 84, 56, 76, 50, 84, 111, 33, 119, 105, 69, 75, 95, 152, 14, 20, 55, 74, 110, 33, 153, 140, 14, 35, 56, 132, 174, 20, 35, 90

Offset: 1

Eric W. Weisstein, Mar 01 2009

nonn

			{20, 14}, {35, 14}, {35, 20}, {54, 44}, {65, 39}, {69, 45}, {77, 21}, {84, 34}, {84, 50}, {95, 75}, ...

Eric Weisstein's World of Mathematics, Visible Point

Cf. A157426, A157428, A157429.

Cf. A325602, A325603, A325604, A325605, A325606, A325607.

A325605 Lower left-hand y-coordinate for 3 X 3 invisible forest with 0 < x < y.

Original entry on oeis.org

1308, 2000, 2330, 2714, 3128, 2540, 2924, 3080, 3484, 3794, 3730, 4654, 4730, 4234, 4640, 4718, 5300, 5564, 5654, 4928, 5704, 5654, 5718, 4598, 5654, 4640, 5642, 6200, 4640, 5150, 4598, 6094, 5984, 5984, 6408, 6460, 4674, 4794, 6104, 5620

Offset: 1

Benjamin Hutz, May 31 2019

nonn

These are 3 X 3 rectangles of lattice points not visible along straight lines of sight from the origin. The sequence is ordered by Euclidean distance from (0,0).

			1308 is a term because (1274, 1308) is the lower left-hand coordinate of a 3 X 3 invisible forest, i.e., gcd(1274+i, 1308+j) > 1 for 0 <= i <= 2 and 0 <= j <= 2.
Other such coordinates are (1884, 2000), (1924, 2330), (1448, 2714), (594, 3128), (2254, 2540), (2364, 2924), (2210, 3080), (1598, 3484), (1000, 3794).

E. Goins, P. Harris, B. Kubik, A. Mbirika, Lattice Point Visibility on Generalized Lines of Sight, arXiv:1710.04554 [math.NT], 2017; Amer. Math. Monthly 125 (2018) 593-601.
F. Herzog, B. M. Stewart, Patterns of Visible and Nonvisible Lattice Points, Amer. Math. Monthly 78 (1971) 487-496.
S. Laishram, F. Luca, Rectangles Of Nonvisible Lattice Points, J. Int. Seq. 18 (2015) 15.10.8.

Cf. A157426, A157427, A157428, A157429.

Cf. A325602, A325603, A325604, A325606, A325607.

A325607 Lower left-hand y-coordinate for 4 X 4 invisible forest with 0 < x < y.

Original entry on oeis.org

10199370, 13125369, 13458288, 21374352, 22339875, 29634813, 39738060, 32719728, 42182412, 47211294, 51258282, 53418859, 43209969, 55756413, 48330114, 49362234, 62965329, 52850433, 72584082, 73167345, 73891893, 71088744, 61716444, 66526029, 69342999, 80514873, 71127132

Offset: 1

Benjamin Hutz, May 31 2019

nonn

These are 4 X 4 rectangles of lattice points not visible along straight lines of sight from the origin. The sequence is ordered by Euclidean distance from (0,0).

			(7247643, 10199370), (6349914, 13125369), (13449225, 13458288), (3268473, 21374352), (16799913, 22339875)

E. Goins, P. Harris, B. Kubik, A. Mbirika, Lattice Point Visibility on Generalized Lines of Sight, arXiv:1710.04554 [math.NT], 2017; Amer. Math. Monthly 125 (2018) 593-601.
F. Herzog, B. M. Stewart, Patterns of Visible and Nonvisible Lattice Points, Amer. Math. Monthly 78 (1971) 487-496
S. Laishram, F. Luca, Rectangles Of Nonvisible Lattice Points, J. Int. Seq. 18 (2015) 15.10.8.

Cf. A157426, A157427, A157428, A157429.

Cf. A325602, A325603, A325604, A325605, A325606.

a(6)-a(27) from Giovanni Resta, Jun 05 2019

A332582 Label the cells of the infinite 2D square lattice with the square spiral (or Ulam spiral), starting with 1 at the center; sequence lists primes that are visible from square 1.

Original entry on oeis.org

2, 3, 5, 7, 29, 41, 47, 83, 89, 97, 103, 107, 109, 113, 173, 179, 181, 191, 193, 199, 223, 293, 311, 317, 347, 353, 359, 443, 449, 457, 461, 467, 479, 487, 491, 499, 503, 509, 521, 523, 631, 641, 643, 647, 653, 659, 661, 673, 683, 691, 701, 709, 719, 727, 857, 863, 887, 929, 947, 953, 1091

Offset: 1

Scott R. Shannon, Feb 17 2020

nonn

Any grid point with relative coordinates (x,y) from the central grid point, which is numbered 1, and where the greatest common divisor (gcd) of |x| and |y| equals 1 will be visible from the central point. Grid points where gcd(|x|,|y|) > 1 will have another point directly between it and the central point and will thus not be visible. In an infinite 2D square lattice the ratio of visible grid points to all points is 6/Pi^2, approximately 0.608, the same as the probability of two random numbers being relative prime.
For a square spiral of size 10001 X 10001, slightly over 100 million numbers, a total of 60803664 numbers are visible, of which 2155170 are prime. The total number of primes in the same range is 5762536, giving a ratio of visible primes to all primes of about 0.374. This is significantly lower than the ratio for all numbers of 0.608, indicating a prime is more likely to be hidden from the origin than a random number.
Primes p such that A174344(p) and A268038(p) are coprime. - Robert Israel, Feb 16 2024

			The 2D grid is shown below. Composite numbers are shown as a '*'. The primes that are blocked from the central 1 square are in parentheses; these all have another composite or prime number directly between their position and the central square.
.
.
    *----*----*--(61)---*--(59)---*----*
                                       |
  (37)---*----*----*----*----*--(31)   *
    |                             |    |
    *  (17)---*----*----*--(13)   *    *
    |    |                   |    |    |
    *    *    5----*----3    *   29    *
    |    |    |         |    |    |    |
    *  (19)   *    1----2  (11)   *  (53)
    |    |    |              |    |    |
   41    *    7----*----*----*    *    *
    |    |                        |    |
    *    *----*--(23)---*----*----*    *
    |                                  |
  (43)---*----*----*---47----*----*----*
.
.
a(1) = 2 to a(4) = 7 are all primes adjacent to the central 1 point, thus all are visible from that square.
a(5) = 29 as primes 11, 13, 17, 19, 23 are blocked from the central 1 point by points numbered 2, 3, 5, 6, 8 respectively.

Robert Israel, Table of n, a(n) for n = 1..10000
Scott R. Shannon, Image showing the visible primes from point 1 for the first 100000 grid points. The visible primes are highlighted in yellow while the hidden primes are highlighted in red. The central 1 square is white while the nonprimes are gray. Zoom into the image to see the grid point numbers.
Eric Weisstein's World of Mathematics, Visible Point.
Wikipedia, Ulam Spiral.

Cf. A000040, A063826, A157426, A157428, A174344, A268038, A325604, A325606, A331400, A332583.

x:= 0: y:= 0: R:= NULL: count:= 0:
for i from 2 while count < 100 do
  if x >= y then
    if x < -y + 1 then x:= x+1
    elif x > y then y:= y+1
    else x:= x-1
    fi
  elif x <= -y then y:= y-1
    else x:= x-1
  fi;
  if isprime(i) and igcd(abs(x),abs(y))=1 then R:= R,i; count:= count+1 fi
od:
R; # Robert Israel, Feb 16 2024

cp's OEIS Frontend

A157428 Lower left-hand x-coordinates of invisible 2 X 2 squares with 0 < y < x.

Views

Author

Keywords

Examples

Links

Crossrefs

A157426 Lower left-hand x-coordinates of invisible 1x1 squares with 0 < y < x.

Views

Author

Keywords

Examples

Links

Crossrefs

A325604 Lower left-hand x-coordinate for 3 X 3 invisible forest with 0 < x < y.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A331400 The grid points visible from the central point of an infinite 2D square lattice where all grid points are numbered as in the Ulam spiral.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A325602 Lower left-hand x-coordinate for 2 X 2 invisible forest with 0 < x < y.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Python

A325603 Lower left-hand y-coordinate for 2 X 2 invisible forest with 0 < x < y.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

SageMath

A157427 Lower left-hand y-coordinates of invisible 1x1 squares with 0 < y < x.

Views

Author

Keywords

Examples

Links

Crossrefs

A325605 Lower left-hand y-coordinate for 3 X 3 invisible forest with 0 < x < y.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A325607 Lower left-hand y-coordinate for 4 X 4 invisible forest with 0 < x < y.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Extensions

A332582 Label the cells of the infinite 2D square lattice with the square spiral (or Ulam spiral), starting with 1 at the center; sequence lists primes that are visible from square 1.

Views

Author

Keywords

Comments