A325603 -id:A325603 - 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.

A325606 Lower left-hand x-coordinate for 4 X 4 invisible forest with 0 < x < y.

Original entry on oeis.org

7247643, 6349914, 13449225, 3268473, 16799913, 12209988, 7676094, 25869732, 27330093, 19113828, 3689958, 8579373, 35193585, 10557390, 31662993, 40976505, 20262723, 47516988, 12932424, 10792494, 13027728, 24801060, 44040840, 37959282, 42196362, 16439577, 43732038

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, A325607.

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

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)

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

cp's OEIS Frontend

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

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

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

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A325606 Lower left-hand x-coordinate for 4 X 4 invisible forest with 0 < x < y.

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

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

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

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

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