A046080 -id:A046080 - cp's OEIS frontend

A328712 Number of non-primitive Pythagorean triples with hypotenuse n.

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 2, 1, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 0, 2, 0, 0, 1, 0, 1, 0, 0, 0, 1, 2, 0, 0, 1, 0, 1, 0, 1, 0, 0, 2, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 0, 2, 0, 1

Offset: 1

Rui Lin, Oct 26 2019

nonn

Pythagorean triple including primitive ones and non-primitive ones. For a certain n, it may be the hypotenuse in either primitive Pythagorean triple, or non-primitive Pythagorean triple, or both.

This sequence is the count of n as hypotenuse in non-primitive Pythagorean triple.

			n=5 as hypotenuse in only one primitive Pythagorean triple, (3,4,5); so a(5)=0.
n=10 as hypotenuse in only one non-primitive Pythagorean triple, (6,8,10); so a(10)=1.
n=25 as hypotenuse in one primitive Pythagorean triple (7,24,25) and in one non-primitive Pythagorean triple (15,20,25); so a(25)=1.

A. Beiler, Recreations in the Theory of Numbers. New York: Dover Publications, pp. 116-117, 1966.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A046080, A024362.

f:= proc(n) local R;
if isprime(n) then return 0 fi;
  R:= map(t -> subs(t,[x,y]),[isolve(x^2+y^2=n^2)]);
  nops(select(t -> t[1]>=1 and t[2]>=t[1] and igcd(t[1],t[2])>1, R))
end proc:
map(f, [$1..100]); # Robert Israel, Oct 31 2019

a[n_] := Module[{R, x, y}, If[PrimeQ[n], 0, R = Solve[GCD[x, y] > 1 && x >= 1 && y >= x && x^2 + y^2 == n^2, {x, y}, Integers]; Length[R]]];
Array[a, 102] (* Jean-François Alcover, Jun 20 2020, after Maple *)

a(n) = A046080(n) - A024362(n).

A046110 Number of lattice points on circumference of a circle of radius (2n+1)/2 with center at (1/2,0).

Original entry on oeis.org

2, 2, 6, 2, 2, 2, 6, 6, 6, 2, 2, 2, 10, 2, 6, 2, 2, 6, 6, 6, 6, 2, 6, 2, 2, 6, 6, 6, 2, 2, 6, 2, 18, 2, 2, 2, 6, 10, 2, 2, 2, 2, 18, 6, 6, 6, 2, 6, 6, 2, 6, 2, 6, 2, 6, 6, 6, 6, 6, 6, 2, 6, 14, 2, 2, 2, 2, 6, 6, 2, 2, 6, 18, 2, 6, 2, 6, 6, 6, 6, 2, 2, 6, 2, 10, 2, 6, 10, 2, 2, 6, 6, 18, 6, 2, 2, 6, 18

Offset: 0

Eric W. Weisstein

nonn

Eric Weisstein's World of Mathematics, Circle Lattice Points.

Cf. A046109.

a(n) = 4 * A046080(2*n+1) + 2. - Sean A. Irvine, Apr 02 2021

A063468 Number of Pythagorean triples in the range [1..n], i.e., the number of integer solutions to x^2 + y^2 = z^2 with 1 <= x,y,z <= n.

Original entry on oeis.org

0, 0, 0, 0, 2, 2, 2, 2, 2, 4, 4, 4, 6, 6, 8, 8, 10, 10, 10, 12, 12, 12, 12, 12, 16, 18, 18, 18, 20, 22, 22, 22, 22, 24, 26, 26, 28, 28, 30, 32, 34, 34, 34, 34, 36, 36, 36, 36, 36, 40, 42, 44, 46, 46, 48, 48, 48, 50, 50, 52, 54, 54, 54, 54, 62, 62, 62, 64, 64, 66, 66, 66, 68, 70

Offset: 1

Ahmed Fares (ahmedfares(AT)my-deja.com), Jul 27 2001

nonn

			For n = 5 the Pythagorean triples are (3, 4, 5) and (4, 3, 5), so a (5) = 2.
For n = 10 the Pythagorean triples are (3, 4, 5), (4, 3, 5), (6, 8, 10) and (8, 6, 10), so a(10) = 4.
For n = 17 the Pythagorean triples are (3, 4, 5), (4, 5, 3), (5, 12, 13), (12, 5, 13), (6, 8, 10), (8, 6, 10), (8, 15, 17), (15, 8, 17), (9, 12, 15) and (12, 9, 15), so a(17) = 10.

Marius A. Burtea, Table of n, a(n) for n = 1..1000

Cf. A062775, A211422.

a(n) = 2*partial sums of A046080(n).

[#[: x in [1..n], y in [1..n]| IsSquare(x^2+y^2) and Floor(Sqrt(x^2+y^2)) le n]:n in [1..74]]; // Marius A. Burtea, Jan 22 2020

nq[n_] := SquaresR[2, n^2]/4 - 1; Accumulate@ Array[nq, 80] (* Giovanni Resta, Jan 23 2020 *)

Corrected and extended by Vladeta Jovovic, Jul 28 2001

A108707 Minimum side in Pythagorean triangles with hypotenuse of n.

Original entry on oeis.org

0, 0, 0, 0, 3, 0, 0, 0, 0, 6, 0, 0, 5, 0, 9, 0, 8, 0, 0, 12, 0, 0, 0, 0, 7, 10, 0, 0, 20, 18, 0, 0, 0, 16, 21, 0, 12, 0, 15, 24, 9, 0, 0, 0, 27, 0, 0, 0, 0, 14, 24, 20, 28, 0, 33, 0, 0, 40, 0, 36, 11, 0, 0, 0, 16, 0, 0, 32, 0, 42, 0, 0, 48, 24, 21, 0, 0, 30, 0, 48, 0, 18, 0, 0, 13, 0, 60, 0, 39, 54

Offset: 1

Sébastien Dumortier, Jun 20 2005

nonn

			a(5) = 3 as the right triangle with sides (3, 4, 5) has hypotenuse n = 5 smallest side a(5) = 3. This is the smallest side a right triangle with integer sides and hypotenuse 5 can have. - _David A. Corneth_, Apr 10 2021

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A083025, A046083, A046084, A009000, A009003, A108708.

A046080 gives the number of Pythagorean triangles with hypotenuse n.

f[n_]:=Block[{k=n-1,m=Sqrt[n/2],a},While[k>m&&!IntegerQ[(a=Sqrt[n^2-k^2])],k--];If[k<=m,0,a]];Table[f[n],{n,90}]

first(n) = {my(lh = List(), res = vector(n, i, oo)); for(u = 2, sqrtint(n), for(v = 1, u, if (u^2+v^2 > n, break); if ((gcd(u, v) == 1) && (0 != (u-v)%2), for (i = 1, n, if (i*(u^2+v^2) > n, break); listput(lh, i*(u^2+v^2)); res[i*(u^2+v^2)] = vecmin([res[i*(u^2+v^2)], i*(u^2 - v^2), i*2*u*v]))))); for(i = 1, n, if(res[i] == oo, res[i] = 0)); res } \\ David A. Corneth, Apr 10 2021, adapted from A009000

Extended by Ray Chandler, Dec 20 2011

A108708 Maximum side length in Pythagorean triangles with hypotenuse n.

Original entry on oeis.org

0, 0, 0, 0, 4, 0, 0, 0, 0, 8, 0, 0, 12, 0, 12, 0, 15, 0, 0, 16, 0, 0, 0, 0, 24, 24, 0, 0, 21, 24, 0, 0, 0, 30, 28, 0, 35, 0, 36, 32, 40, 0, 0, 0, 36, 0, 0, 0, 0, 48, 45, 48, 45, 0, 44, 0, 0, 42, 0, 48, 60, 0, 0, 0, 63, 0, 0, 60, 0, 56, 0, 0, 55, 70, 72, 0, 0, 72, 0, 64, 0, 80, 0, 0, 84, 0, 63, 0

Offset: 1

Sébastien Dumortier, Jun 20 2005

nonn

			a(5) is 4 as the maximum side (other than the hypotenuse) a right triangle with integer sides and hypotenuse 5 can have.

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A083025, A046083, A046084, A009000, A009003, A108707.

A046080 gives the number of Pythagorean triangles with hypotenuse n.

f[n_] := Block[{k = n - 1, m = Sqrt[n/2]}, While[k > m && !IntegerQ[Sqrt[n^2 - k^2]], k-- ]; If[k <= m, 0, k]]; Table[ f[n], {n, 90}] (* Robert G. Wilson v, Jun 21 2005 *)

first(n) = {my(lh = List(), res = vector(n)); for(u = 2, sqrtint(n), for(v = 1, u, if (u^2+v^2 > n, break); if ((gcd(u, v) == 1) && (0 != (u-v)%2), for (i = 1, n, if (i*(u^2+v^2) > n, break); listput(lh, i*(u^2+v^2)); res[i*(u^2+v^2)] = max(res[i*(u^2+v^2)], max(i*(u^2 - v^2), i*2*u*v)); ); ); ); ); for(i = 1, n, if(res[i] == oo, res[i] = 0)); res } \\ David A. Corneth, Apr 10 2021, adapted from A009000

More terms from Robert G. Wilson v, Jun 21 2005

A256452 Number of integer solutions to n^2 = x^2 + y^2 with x>0, y>=0.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 1, 1, 3, 1, 1, 3, 1, 3, 1, 3, 1, 1, 3, 1, 1, 1, 1, 5, 3, 1, 1, 3, 3, 1, 1, 1, 3, 3, 1, 3, 1, 3, 3, 3, 1, 1, 1, 3, 1, 1, 1, 1, 5, 3, 3, 3, 1, 3, 1, 1, 3, 1, 3, 3, 1, 1, 1, 9, 1, 1, 3, 1, 3, 1, 1, 3, 3, 5, 1, 1, 3, 1, 3, 1, 3, 1, 1, 9, 1, 3

Offset: 1

Michael Somos, Mar 29 2015

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A046080, A046109, A002654.

a:= n-> add(`if`(d::odd, (-1)^((d-1)/2), 0), d=numtheory[divisors](n^2)): seq(a(n), n=1..100);  # Ridouane Oudra, Aug 18 2024

a[ n_] := Sum[ Mod[ Length@Divisors[n^2 - k^2], 2], {k, n}];
a[ n_] := Length @ FindInstance[ n^2 == x^2 + y^2 && x > 0 && y >= 0, {x, y}, Integers, 10^9]; (* Michael Somos, Aug 15 2016 *)
f[p_, e_] := If[Mod[p, 4] == 1, 2*e + 1, 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 12 2020 *)

{a(n) = sum(k=1, n, issquare(n^2 - k^2))};

Multiplicative with a(p^e) = 2*e + 1 if p == 1 (mod 4), otherwise a(p^e) = 1.
a(n) = 1 + 2*A046080(n) if n>0.
a(n) = A046109(n)/4 for n > 0. - Hugo Pfoertner, Sep 21 2023
a(n) = A002654(n^2). - Ridouane Oudra, Aug 18 2024

A349538 The number of pseudo-Pythagorean triples (which allow negative or 0 sides) on a 2D lattice that are on or inside a circle of radius n.

Original entry on oeis.org

1, 5, 9, 13, 17, 29, 33, 37, 41, 45, 57, 61, 65, 77, 81, 93, 97, 109, 113, 117, 129, 133, 137, 141, 145, 165, 177, 181, 185, 197, 209, 213, 217, 221, 233, 245, 249, 261, 265, 277, 289, 301, 305, 309, 313, 325, 329, 333, 337, 341, 361, 373, 385, 397, 401, 413, 417, 421, 433, 437, 449

Offset: 0

Alexander Kritov, Nov 21 2021

nonn

Consider a 2D lattice, where the Cartesian coordinates x and y are legs of the Pythagorean triangle. Thus the notion of Pythagorean triple is extended to the cases when sides x, y are in Z (i.e., sides also include negative integers and zero). The sequence gives the number of such triples on or inside a circle of radius n.

Partial sums of A046109.

			Sides (coordinates)                                                       a(n)
------------------------------------------------------------------------------
(0,0)                                                                       1
(-1,0)(0,-1)(0,1)(1,0)                                                      5
(-2,0)(0,-2)(0,2)(2,0)                                                      9
(-3,0)(0,-3)(0,3)(3,0)                                                     13
(-4,0)(0,-4)(0,4)(4,0)                                                     17
(-5,0)(-4,-3)(-4,3)(-3,-4)(-3,4)(0,-5)(0,5)(3,-4)(3,4)(4,-3)(4,3)(5,0)     29
(-6,0)(0,-6)(0,6)(6,0)                                                     33
(-7,0)(0,-7)(0,7)(7,0)                                                     37
(-8,0)(0,-8)(0,8)(8,0)                                                     41
(-9,0)(0,-9)(0,9)(9,0)                                                     45
(-10,0)(-8,-6)(-8,6)(-6,-8)(-6,8)(0,-10)(0,10)(6,-8)(6,8)(8,-6)(8,6)(10,0) 57
(-11,0)(0,-11)(0,11)(11,0)                                                 61
(-12,0)(0,-12)(0,12)(12,0)                                                 65

Alexander Kritov, Source code

Cf. A046080, A211432, A046109 (first differences), A349536 (in 1/8 sector).

C
```
/* See links */
```

f(n) = if(n==0, return(1)); my(f=factor(n)); 4*prod(i=1, #f~, if(f[i, 1]%4==1, 2*f[i, 2]+1, 1)); \\ A046109
a(n) = sum(k=0, n, f(k)); \\ Michel Marcus, Nov 27 2021

a(n) = (A211432(n) + 1)/2.

a(n) = a(n-1) + 4 + 8*A046080(n).

A056138 Number of ways in which n can be the shorter leg (shortest side) of an integer-sided right triangle.

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 3, 2, 1, 2, 1, 3, 3, 1, 1, 4, 2, 1, 3, 3, 1, 3, 1, 3, 4, 1, 3, 5, 1, 1, 4, 5, 1, 3, 1, 3, 5, 1, 1, 7, 2, 2, 4, 3, 1, 3, 3, 5, 4, 1, 1, 9, 1, 1, 5, 4, 4, 4, 1, 3, 4, 3, 1

Offset: 1

Henry Bottomley, Jun 15 2000

nonn

Cf. A009004, A055525, A055527.

a(n)=my(b);sum(c=n+2,n^2\2+1,issquare(c^2-n^2,&b) && nCharles R Greathouse IV, Jul 07 2013

a(n) = A046079(n) - A056137(n) = A046081(n) - A046080(n) - A056137(n).

A063669 Hypotenuses of reciprocal Pythagorean triangles: number of solutions to 1/(12n)^2 = 1/b^2 + 1/c^2 [with b >= c > 0]; also number of values of A020885 (with repetitions) which divide n.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 1, 1, 3, 1, 1, 1, 2, 2, 1, 1, 1, 1, 3, 1, 1, 1, 1, 2, 1, 1, 2, 1, 4, 1, 1, 1, 1, 4, 1, 1, 1, 1, 3, 1, 2, 1, 1, 2, 1, 1, 1, 1, 3, 1, 1, 1, 1, 3, 2, 1, 1, 1, 4, 1, 1, 1, 1, 2, 1, 1, 1, 1, 6, 1, 1, 1, 1, 2, 1, 1, 1, 1, 3, 1, 1, 1, 3, 2, 1, 1, 1, 1, 4, 2, 1, 1, 1, 2, 1, 1, 2, 1, 3, 1, 1, 1, 1, 5

Offset: 1

Henry Bottomley, Jul 28 2001

nonn

Primitive reciprocal Pythagorean triangles 1/a^2 = 1/b^2 + 1/c^2 have a=fg, b=ef, c=eg where e^2 = f^2 + g^2; i.e., e,f,g represent the sides of primitive Pythagorean triangles. But the product of the two legs of primitive Pythagorean triangles are multiples of 12 and so the reciprocal of hypotenuses of reciprocal Pythagorean triangles are always multiples of 12 (A008594).

			a(1)=1 since 1/(12*1)^2 = 1/12^2 = 1/15^2 + 1/20^2;
a(70)=6 since 1/(12*70)^2 = 1/840^2 = 1/875^2 + 1/3000^2 = 1/888^2 + 1/2590^2 = 1/910^2 + 1/2184^2 = 1/952^2 + 1/1785^2 = 1/1050^2 + 1/1400^2 = 1/1160^2 + 1/1218^2.
Looking at A020885, 1 is divisible by 1, while 70 is divisible by 1, 5, 10, 14, 35 and again 35.

Cf. A046080, A063664, A063665, A063014.

A342154 Number of partitions of n^5 into two positive squares.

Original entry on oeis.org

0, 0, 1, 0, 0, 3, 0, 0, 1, 0, 3, 0, 0, 3, 0, 0, 0, 3, 1, 0, 3, 0, 0, 0, 0, 5, 3, 0, 0, 3, 0, 0, 1, 0, 3, 0, 0, 3, 0, 0, 3, 3, 0, 0, 0, 3, 0, 0, 0, 0, 6, 0, 3, 3, 0, 0, 0, 0, 3, 0, 0, 3, 0, 0, 0, 18, 0, 0, 3, 0, 0, 0, 1, 3, 3, 0, 0, 0, 0, 0, 3, 0, 3, 0, 0, 18, 0, 0, 0, 3, 3, 0, 0, 0, 0, 0, 0, 3, 1, 0, 5, 3, 0, 0, 3, 0

Offset: 0

Seiichi Manyama, Mar 02 2021

a(n) > 0 if and only if n is in A000404. - Robert Israel, Mar 03 2021

			2^5 = 32 = 4^2 + 4^2. So a(2) = 1.
5^5 = 3125 = 10^2 + 55^2 = 25^2 + 50^2 = 38^2 + 41^2. So a(5) = 3.

Robert Israel, Table of n, a(n) for n = 0..3050
Index entries for sequences related to sums of squares

Cf. A000404, A000584, A025426, A046080, A084888.

f:= proc(n) local x,y,S;
      S:= map(t -> subs(t,[x,y]),[isolve(x^2+y^2=n^5)]);
      nops(select(t -> t[1] >= t[2] and t[2] > 0, S))
end proc:
map(f, [$0..200]); # Robert Israel, Mar 03 2021

a(n) = my(cnt=0, m=n^5); for(k=1, sqrt(m/2), l=m-k*k; if(l>0&&issquare(l), cnt++)); cnt;

a(n) = A025426(A000584(n)).

cp's OEIS Frontend

A328712 Number of non-primitive Pythagorean triples with hypotenuse n.

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A046110 Number of lattice points on circumference of a circle of radius (2n+1)/2 with center at (1/2,0).

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A063468 Number of Pythagorean triples in the range [1..n], i.e., the number of integer solutions to x^2 + y^2 = z^2 with 1 <= x,y,z <= n.

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Magma

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A108707 Minimum side in Pythagorean triangles with hypotenuse of n.

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Mathematica

PARI

Extensions

A108708 Maximum side length in Pythagorean triangles with hypotenuse n.

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Mathematica

PARI

Extensions

A256452 Number of integer solutions to n^2 = x^2 + y^2 with x>0, y>=0.

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Maple

Mathematica

PARI

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A349538 The number of pseudo-Pythagorean triples (which allow negative or 0 sides) on a 2D lattice that are on or inside a circle of radius n.

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C

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A056138 Number of ways in which n can be the shorter leg (shortest side) of an integer-sided right triangle.

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