A046109 -id:A046109 - cp's OEIS frontend

A290502 Hypotenuses for which there exist exactly 14 distinct integer triangles.

6103515625, 12207031250, 18310546875, 24414062500, 36621093750, 42724609375, 48828125000, 54931640625, 67138671875, 73242187500, 85449218750, 97656250000, 109863281250, 115966796875, 128173828125, 134277343750, 140380859375, 146484375000, 164794921875

Offset: 1

Hamdi Sahloul, Aug 04 2017

nonn

Numbers whose square is decomposable in 14 different ways into the sum of two nonzero squares: these are those with only one prime divisor of the form 4k+1 with multiplicity fourteen.

			a(1) = 6103515625 = 5^14, a(5) = 36621093750 = 2*3*5^14, a(101) = 1171875000000 = 2^6*3*5^14.

Ray Chandler, Table of n, a(n) for n = 1..10000 (first 1000 terms from Hamdi Sahloul)

Cf. A002144, A006339, A046080, A046109, A083025.

Cf. A004144 (0), A084645 (1), A084646 (2), A084647 (3), A084648 (4), A084649 (5), A097219 (6), A097101 (7), A290499 (8), A290500 (9), A097225 (10), A290501 (11), A097226 (12), A097102 (13), A290503 (15), A097238 (16), A097239 (17), A290504 (18), A290505 (19), A097103 (22), A097244 (31), A097245 (37), A097282 (40), A097626 (67).

r[a_]:={b, c}/.{ToRules[Reduce[0Vincenzo Librandi, Mar 01 2016 *)

Terms are obtained by the product A004144(k)*A002144(p)^14 for k, p > 0 ordered by increasing values.

A290503 Hypotenuses for which there exist exactly 15 distinct integer triangles.

Original entry on oeis.org

30517578125, 61035156250, 91552734375, 122070312500, 183105468750, 213623046875, 244140625000, 274658203125, 335693359375, 366210937500, 427246093750, 488281250000, 549316406250, 579833984375, 640869140625, 671386718750, 701904296875, 732421875000

Offset: 1

Hamdi Sahloul, Aug 04 2017

nonn

Numbers whose square is decomposable in 15 different ways into the sum of two nonzero squares: these are those with only one prime divisor of the form 4k+1 with multiplicity fifteen.

			a(1) = 30517578125 = 5^15, a(5) = 183105468750 = 2*3*5^15, a(101) = 5859375000000 = 2^6*3*5^15.

Ray Chandler, Table of n, a(n) for n = 1..10000 (first 1000 terms from Hamdi Sahloul)

Cf. A002144, A006339, A046080, A046109, A083025.

Cf. A004144 (0), A084645 (1), A084646 (2), A084647 (3), A084648 (4), A084649 (5), A097219 (6), A097101 (7), A290499 (8), A290500 (9), A097225 (10), A290501 (11), A097226 (12), A097102 (13), A290502 (14), A097238 (16), A097239 (17), A290504 (18), A290505 (19), A097103 (22), A097244 (31), A097245 (37), A097282 (40), A097626 (67).

r[a_]:={b, c}/.{ToRules[Reduce[0Vincenzo Librandi, Mar 01 2016 *)

Terms are obtained by the product A004144(k)*A002144(p)^15 for k, p > 0 ordered by increasing values.

A290504 Hypotenuses for which there exist exactly 18 distinct integer triangles.

Original entry on oeis.org

3814697265625, 7629394531250, 11444091796875, 15258789062500, 22888183593750, 26702880859375, 30517578125000, 34332275390625, 41961669921875, 45776367187500, 53405761718750, 61035156250000, 68664550781250, 72479248046875, 80108642578125, 83923339843750

Offset: 1

Hamdi Sahloul, Aug 04 2017

nonn

Numbers whose square is decomposable in 18 different ways into the sum of two nonzero squares: these are those with only one prime divisor of the form 4k+1 with multiplicity eighteen.

			a(1) = 3814697265625 = 5^18, a(5) = 22888183593750 = 2*3*5^18, a(101) = 732421875000000 = 2^6*3*5^18.

Ray Chandler, Table of n, a(n) for n = 1..10000 (first 1000 terms from Hamdi Sahloul)

Cf. A002144, A006339, A046080, A046109, A083025.

Cf. A004144 (0), A084645 (1), A084646 (2), A084647 (3), A084648 (4), A084649 (5), A097219 (6), A097101 (7), A290499 (8), A290500 (9), A097225 (10), A290501 (11), A097226 (12), A097102 (13), A290502 (14), A290503 (15), A097238 (16), A097239 (17), A290505 (19), A097103 (22), A097244 (31), A097245 (37), A097282 (40), A097626 (67).

r[a_]:={b, c}/.{ToRules[Reduce[0Vincenzo Librandi, Mar 01 2016 *)

Terms are obtained by the product A004144(k)*A002144(p)^18 for k, p > 0 ordered by increasing values.

A290505 Hypotenuses for which there exist exactly 19 distinct integer triangles.

Original entry on oeis.org

203125, 265625, 406250, 453125, 531250, 578125, 609375, 640625, 796875, 812500, 828125, 906250, 953125, 1062500, 1140625, 1156250, 1218750, 1281250, 1359375, 1390625, 1421875, 1515625, 1578125, 1593750, 1625000, 1656250, 1703125, 1734375, 1765625, 1812500

Offset: 1

Hamdi Sahloul, Aug 04 2017

nonn

Numbers whose square is decomposable in 19 different ways into the sum of two nonzero squares: these are those with exactly two distinct prime divisors of the form 4k+1 with one, and six respective multiplicities, or with only one prime divisor of this form with multiplicity nineteen.

			a(1) = 203125 = 5^6*13, a(5) = 531250 = 2*5^6*17, a(281) = 12796875 = 3^2*5^6*7*13.

Ray Chandler, Table of n, a(n) for n = 1..10000 (first 1000 terms from Hamdi Sahloul)

Cf. A002144, A006339, A046080, A046109, A083025.

Cf. A004144 (0), A084645 (1), A084646 (2), A084647 (3), A084648 (4), A084649 (5), A097219 (6), A097101 (7), A290499 (8), A290500 (9), A097225 (10), A290501 (11), A097226 (12), A097102 (13), A290502 (14), A290503 (15), A097238 (16), A097239 (17), A290504 (18), A097103 (22), A097244 (31), A097245 (37), A097282 (40), A097626 (67).

r[a_]:={b, c}/.{ToRules[Reduce[0Vincenzo Librandi, Mar 01 2016 *)

Terms are obtained by the products A004144(k)*A002144(p1)*A002144(p2)^6, or A004144(k)*A002144(p1)^19 for k, p1, p2 > 0 ordered by increasing values.

A051132 Number of ordered pairs of integers (x,y) with x^2+y^2 < n^2.

Original entry on oeis.org

0, 1, 9, 25, 45, 69, 109, 145, 193, 249, 305, 373, 437, 517, 609, 697, 793, 889, 1005, 1125, 1245, 1369, 1513, 1649, 1789, 1941, 2109, 2285, 2449, 2617, 2809, 2997, 3205, 3405, 3613, 3841, 4049, 4281, 4509, 4765, 5013, 5249, 5521, 5785, 6073, 6349, 6621

Offset: 0

Jostein Trondal (jostein.trondal(AT)protech.no)

			a(3)=25 from the points of shapes 00 (1), 10 (4), 11 (4), 20 (4), 21 (8), 22 (4).

T. D. Noe, Table of n, a(n) for n = 0..1000
Jianqiang Zhao, The Largest Circle Enclosing n Lattice Points, arXiv:2505.06234 [math.GM], 2025. See p. 18.

Changing "<" to "<=" in the definition gives A000328.

a051132 n = length [(x,y) | x <- [-n..n], y <- [-n..n], x^2 + y^2 < n^2]
-- Reinhard Zumkeller, Jan 23 2012

Table[Sum[SquaresR[2, k], {k, 0, n^2 - 1}], {n, 0, 46}]
a[0]=0;a[n_]:=4*n-3+4Sum[Ceiling[Sqrt[n^2-i^2]]-1,{i,n-1}];Array[a,47,0] (* Giorgos Kalogeropoulos, May 20 2025 *)

from math import isqrt
def A051132(n): return 1+(sum(isqrt(k*((n<<1)-k)-1) for k in range(1,n+1))<<2) if n else 0 # Chai Wah Wu, Feb 12 2025

a(n) = A000328(n) - A046109(n). - Reinhard Zumkeller, Jan 23 2012
Limit_{n->oo} a(n)/n^2 = Pi. - Chai Wah Wu, Feb 12 2025
a(n) = 4*n - 3 + 4 Sum_{i=1..n-1} ceiling(sqrt(n^2 - i^2)) - 1, for n > 0 (see Zhao). - Giorgos Kalogeropoulos, May 20 2025

More terms from James Sellers

A242118 Number of unit squares that intersect the circumference of a circle of radius n centered at (0,0).

Original entry on oeis.org

0, 4, 12, 20, 28, 28, 44, 52, 60, 68, 68, 84, 92, 92, 108, 108, 124, 124, 140, 148, 148, 164, 172, 180, 188, 180, 196, 212, 220, 220, 228, 244, 252, 260, 260, 268, 284, 284, 300, 300, 308, 316, 332, 340, 348, 348, 364, 372, 380, 388, 380

Offset: 0

Kival Ngaokrajang, May 05 2014

nonn

For the points that form the Pythagorean triple (for example see illustration n = 5, on the first quadrant at coordinate (4,3) and (3,4)), the transit of circumference occurs exactly at the corners, therefore there are no additional intersecting squares on the upper or lower rows (diagonally NE & SW directions) of such points.

If the center of the circle is instead chosen at the middle of a square grid centered at (1/2,0), the sequence will be 2*A004767(n-1).

Kival Ngaokrajang, Illustration of initial terms

Cf. A009003, A004767.

a = lambda n: sum(4 for x in range(n) for y in range(n)
                    if x**2 + y**2 < n**2 and (x+1)**2 + (y+1)**2 > n**2)

from sympy import factorint
def a(n):
    r = 1
    for p, e in factorint(n).items():
        if p%4 == 1: r *= 2*e + 1
    return 8*n - 4*r if n > 0 else 0

a(n) = 4*Sum{k=1..n} ceiling(sqrt(n^2 - (k-1)^2)) - floor(sqrt(n^2 - k^2)). - Orson R. L. Peters, Jan 30 2017

a(n) = 8*n - A046109(n) for n > 0. - conjectured by Orson R. L. Peters, Jan 30 2017, proved by Andrey Zabolotskiy, Jan 31 2017

Terms corrected by Orson R. L. Peters, Jan 30 2017

A302996 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals: A(n,k) = [x^(n^2)] theta_3(x)^k, where theta_3() is the Jacobi theta function.

Original entry on oeis.org

1, 1, 0, 1, 2, 0, 1, 4, 2, 0, 1, 6, 4, 2, 0, 1, 8, 6, 4, 2, 0, 1, 10, 24, 30, 4, 2, 0, 1, 12, 90, 104, 6, 12, 2, 0, 1, 14, 252, 250, 24, 30, 4, 2, 0, 1, 16, 574, 876, 730, 248, 30, 4, 2, 0, 1, 18, 1136, 3542, 4092, 1210, 312, 54, 4, 2, 0, 1, 20, 2034, 12112, 18494, 7812, 2250, 456, 6, 4, 2, 0

Offset: 0

Ilya Gutkovskiy, Apr 17 2018

A(n,k) is the number of ordered ways of writing n^2 as a sum of k squares.

			Square array begins:
  1,  1,   1,   1,    1,     1,  ...
  0,  2,   4,   6,    8,    10,  ...
  0,  2,   4,   6,   24,    90,  ...
  0,  2,   4,  30,  104,   250,  ...
  0,  2,   4,   6,   24,   730,  ...
  0,  2,  12,  30,  248,  1210,  ...

Alois P. Heinz, Antidiagonals n = 0..200, flattened
Eric Weisstein's World of Mathematics, Jacobi Theta Functions
Index entries for sequences related to sums of squares

Columns k=0..4,7 give A000007, A040000, A046109, A016725, A267326, A361695.
Main diagonal gives A232173.
Cf. A000122, A122141, A255212, A286815.

b:= proc(n, t) option remember; `if`(n=0, 1, `if`(n<0 or t<1, 0,
      b(n, t-1)+2*add(b(n-j^2, t-1), j=1..isqrt(n))))
    end:
A:= (n, k)-> b(n^2, k):
seq(seq(A(n,d-n), n=0..d), d=0..12);  # Alois P. Heinz, Mar 10 2023

Table[Function[k, SeriesCoefficient[EllipticTheta[3, 0, x]^k, {x, 0, n^2}]][j - n], {j, 0, 11}, {n, 0, j}] // Flatten
Table[Function[k, SeriesCoefficient[Sum[x^i^2, {i, -n, n}]^k, {x, 0, n^2}]][j - n], {j, 0, 11}, {n, 0, j}] // Flatten

A(n,k) = [x^(n^2)] (Sum_{j=-infinity..infinity} x^(j^2))^k.

A291259 Minimum number of points of the square lattice falling strictly inside a circle of radius n.

Original entry on oeis.org

0, 1, 9, 25, 45, 69, 108, 145, 193, 248, 305, 373, 437, 517, 608, 697, 793, 889, 1005, 1124, 1245, 1369, 1510, 1649, 1789, 1941, 2109, 2278, 2449, 2617, 2809, 2997, 3202, 3405, 3613, 3834, 4049, 4281, 4509, 4762, 5013, 5249, 5521, 5785, 6068, 6348, 6621, 6917

Offset: 0

Andres Cicuttin, Aug 21 2017

nonn

Due to the symmetry and periodicity of the square lattice it is sufficient to explore possible circles with center belonging to the triangle with vertices (0,0), (1/2,0), and (1/2,1/2).

The different regions for the centers producing constant numbers of lattice points inside circles of radius n seem to become very complex and irregular as n increases (see density plots in Links).

			From _Arkadiusz Wesolowski_, Dec 18 2017 [Corrected by _Andrey Zabolotskiy_, Feb 19 2018]: (Start)
For a circle centered at the point (x, y) = (1/2, 0) with radius 6, there are 108 lattice points inside the circle.
Possible (but not unique) choices for the centers of the circles for radii up to 20 are given below.
.
.  Poss. center          Points in
.    x      y    Radius  the circle
.  -----  -----  ------  ----------
.    0      0       1         1
.    0      0       2         9
.    0      0       3        25
.    0      0       4        45
.    0      0       5        69
.   1/2     0       6       108
.    0      0       7       145
.    0      0       8       193
.   1/5     0       9       248
.    0      0      10       305
.    0      0      11       373
.    0      0      12       437
.    0      0      13       517
.   1/4     0      14       608
.    0      0      15       697
.    0      0      16       793
.    0      0      17       889
.    0      0      18      1005
.   1/2    1/2     19      1124
.    0      0      20      1245
(End)

Robert G. Wilson v, Table of n, a(n) for n = 0..125
Andres Cicuttin, Plots of regions for centers of circles of several radii n enclosing constant number of lattice points
Eric Weisstein's World of Mathematics, Circle Lattice Points

Cf. A046109, A051132, A295344.

(* A291259: Minimum number of points of the square lattice falling strictly inside a circle of radius n. *)
(* The three vertices of the Explorative Triangle (ET) *)
P1={0,0}; P2={1/2,0}; P3={1/2,1/2};
dd2=SquaredEuclideanDistance;
(* candidatePointQ[p,n] gives True if "p" is a candidate point, and False otherwise. A candidate point is a point belonging to a circle of radius "n" with center in the ET *)
candidatePointQ[p_,n_] := With[{dds={dd2[p,P1],dd2[p,P2],dd2[p,P3]}}, Max[dds]>=n^2>=Min[dds]];
(* Check if point "p" falls inside any circle with radius "n" and center in the ET *)
innerPointQ[p_,n_] := With[{dds={dd2[p,P1],dd2[p,P2],dd2[p,P3]}}, Max[dds]Andres Cicuttin & Andrey Zabolotskiy, Nov 14 2017 *)

a(n) ~ Pi*n^2.

a(n) <= A051132(n). - Joerg Arndt, Oct 03 2017

More terms from Andrey Zabolotskiy, Nov 17 2017

A046112 a(n) is smallest integral radius of circle centered at (0,0) having 8n-4 lattice points on its circumference; a(n)/2 is smallest half-integral radius circle centered at (1/2,0) having 4n-2 lattice points; a(n)/3 is smallest third-integral radius circle centered at (1/3,0) having 2n-1 lattice points.

Original entry on oeis.org

1, 5, 25, 125, 65, 3125, 15625, 325, 390625, 1953125, 1625, 48828125, 4225, 1105, 6103515625, 30517578125, 40625, 21125, 3814697265625, 203125, 95367431640625, 476837158203125, 5525, 11920928955078125, 274625

Offset: 1

Eric W. Weisstein

nonn

Ray Chandler, Table of n, a(n) for n = 1..1439 (a(1440) exceeds 1000 digits).
Eric Weisstein's World of Mathematics, Circle Lattice Points
Eric Weisstein's World of Mathematics, Pythagorean Triple

Except for offset, same as A006339.

Cf. A046109, A046110, A046111, A000328.

A071384 Radii of the circles around (0,0) that contain record numbers of lattice points, rounded up to the next integer.

Original entry on oeis.org

0, 1, 3, 5, 9, 19, 34, 65, 75, 167, 269, 372, 401, 896, 1444, 2002, 2435, 5445, 8779, 12175, 15591, 34862, 56213, 77953, 113501, 231769, 253794, 409231, 567501, 886464, 1687299, 1982193, 3196190, 4432317, 7146896, 13178226, 15980946

Offset: 1

Hugo Pfoertner, May 23 2002

nonn

a(n)^2 = A071383(n) for a(n) = 1, 5, 65, ... .

Ray Chandler, Table of n, a(n) for n = 1..425
Hugo Pfoertner, Construction of A071383, A071384, A071385

Cf. A071383, A071385, A046109.

a(n) = ceiling ( A071383(n)^(1/2) ).

Description clarified by Günter Rote, Sep 13 2023

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A290502 Hypotenuses for which there exist exactly 14 distinct integer triangles.

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A290503 Hypotenuses for which there exist exactly 15 distinct integer triangles.

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A290504 Hypotenuses for which there exist exactly 18 distinct integer triangles.

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A290505 Hypotenuses for which there exist exactly 19 distinct integer triangles.

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A051132 Number of ordered pairs of integers (x,y) with x^2+y^2 < n^2.

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A242118 Number of unit squares that intersect the circumference of a circle of radius n centered at (0,0).

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A302996 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals: A(n,k) = [x^(n^2)] theta_3(x)^k, where theta_3() is the Jacobi theta function.

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