A046080 -id:A046080 - cp's OEIS frontend

A097756 Table read by rows of A054994 ordered by A046080.

1, 5, 25, 125, 65, 625, 3125, 15625, 325, 78125, 390625, 1953125, 1625, 9765625, 48828125, 4225, 244140625, 1105, 8125, 1220703125, 6103515625, 30517578125, 40625, 152587890625, 21125, 762939453125, 3814697265625, 203125

Offset: 1

Ray Chandler, Aug 26 2004

Row n of table is A054994(k) such that A046080(A054994(k)) = n; number of terms in row n is A001055(2n+1).

Column 1 of table gives A006339 (or A046112).

			Table begins:
0: 1,
1: 5,
2: 25,
3: 125,
4: 65,625,
5: 3125,
6: 15625,
7: 325,78125,
8: 390625,
9: 1953125,
10: 1625,9765625,
11: 48828125,
12: 4225,244140625,
13: 1105,8125,1220703125,
14: 6103515625,
15: 30517578125,
16: 40625,152587890625,
17: 21125,762939453125,
18: 3814697265625,
19: 203125,19073486328125,
20: 95367431640625,
...

Ray Chandler, Table of n, a(n) for n = 1..4522, 1431 rows flattened.
Eric Weisstein's World of Mathematics, Pythagorean Triple

Cf. A097751, A097752, A097753, A097754, A097755, A054994.

A083025 Number of primes congruent to 1 modulo 4 dividing n (with multiplicity).

Original entry on oeis.org

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

Offset: 1

Reinhard Zumkeller, Oct 29 2001

David A. Cox, "Primes of the Form x^2 + n y^2", Wiley, 1989, p. 61.

T. D. Noe, Table of n, a(n) for n = 1..10000

First differs from A046080 at n=65.

Cf. A001222, A007814, A027746, A065339 (== 3 (mod 4)), A378879 (=2,3 (mod 4)), A005089 (without multiplicity).

a083025 1 = 0
a083025 n = length [x | x <- a027746_row n, mod x 4 == 1]
-- Reinhard Zumkeller, Jan 10 2012

A083025 := proc(n)
    a := 0 ;
    for f in ifactors(n)[2] do
        if op(1,f) mod 4 = 1 then
            a := a+op(2,f) ;
        end if;
    end do:
    a ;
end proc: # R. J. Mathar, Dec 16 2011

f[n_]:=Plus@@Last/@Select[If[n==1,{},FactorInteger[n]],Mod[#[[1]],4]==1&]; Table[f[n],{n,100}] (* Ray Chandler, Dec 18 2011 *)

A083025(n)=sum(i=1,#n=factor(n)~,if(n[1,i]%4==1,n[2,i]))  \\ M. F. Hasler, Apr 16 2012

a(n) = A001222(n) - A007814(n) - A065339(n).

Totally additive with a(2) = 0, a(p) = 1 if p == 1 (mod 4), and a(p) = 0 if p == 3 (mod 4). - Amiram Eldar, Jun 17 2024

A084645 Hypotenuses for which there exists a unique integer-sided right triangle.

Original entry on oeis.org

5, 10, 13, 15, 17, 20, 26, 29, 30, 34, 35, 37, 39, 40, 41, 45, 51, 52, 53, 55, 58, 60, 61, 68, 70, 73, 74, 78, 80, 82, 87, 89, 90, 91, 95, 97, 101, 102, 104, 105, 106, 109, 110, 111, 113, 115, 116, 117, 119, 120, 122, 123, 135, 136, 137, 140, 143, 146, 148, 149

Offset: 1

Eric W. Weisstein, Jun 01 2003

nonn

Numbers whose square is uniquely decomposable into the sum of two nonzero squares: these are those numbers with exactly one prime divisor of the form 4k+1 with multiplicity one. - Jean-Christophe Hervé, Nov 11 2013

Ray Chandler, Table of n, a(n) for n = 1..10000 (first 1000 terms from Zak Seidov)
Eric Weisstein's World of Mathematics, Pythagorean Triple

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

Cf. A004144 (0), 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), A290505 (19), A097103 (22), A097244 (31), A097245 (37), A097282 (40), A097626 (67).

r[a_] := {b, c} /. {ToRules[ Reduce[0 < b < c && a^2 == b^2 + c^2, {b, c}, Integers]]}; Select[ Range[150], Length[r[#]] == 1 &] (* Jean-François Alcover, Oct 22 2012 *)

is_a084645(n) = #qfbsolve(Qfb(1,0,1),n^2,3)==3 \\ Hugo Pfoertner, Sep 28 2024

Terms are obtained by the products A004144(k)*A002144(p) for k, p > 0, ordered by increasing values. - Jean-Christophe Hervé, Nov 12 2013

A046080(a(n)) = 1, A046109(a(n)) = 12. - Jean-Christophe Hervé, Dec 01 2013

A046079 Number of Pythagorean triangles with leg n.

Original entry on oeis.org

0, 0, 1, 1, 1, 1, 1, 2, 2, 1, 1, 4, 1, 1, 4, 3, 1, 2, 1, 4, 4, 1, 1, 7, 2, 1, 3, 4, 1, 4, 1, 4, 4, 1, 4, 7, 1, 1, 4, 7, 1, 4, 1, 4, 7, 1, 1, 10, 2, 2, 4, 4, 1, 3, 4, 7, 4, 1, 1, 13, 1, 1, 7, 5, 4, 4, 1, 4, 4, 4, 1, 12, 1, 1, 7, 4, 4, 4, 1, 10, 4, 1, 1, 13, 4, 1, 4, 7, 1, 7, 4, 4, 4, 1, 4, 13, 1, 2, 7

Offset: 1

Eric W. Weisstein

Number of ways in which n can be the leg (other than the hypotenuse) of a primitive or nonprimitive right triangle.
Number of ways that 2/n can be written as a sum of exactly two distinct unit fractions. For every solution to 2/n = 1/x + 1/y, x < y, the Pythagorean triple is (n, y-x, x+y-n). - T. D. Noe, Sep 11 2002
For n>2, the positions of the ones in this sequence correspond to the prime numbers and their doubles, A001751. - Ant King, Jan 29 2011
Let L = length of longest leg, H = hypotenuse. For odd n: L =(n^2-1)/2 and H = L+1.  For even n,  L = (n^2-4)/4 and H = L+2. - Richard R. Forberg, May 31 2013
Or number of ways n^2 can be written as the difference of two positive squares: a(3) = 1: 3^2 = 5^2-4^2; a(8) = 2: 8^2 = 10^2-6^2 = 17^2-15^2; a(16) = 3: 16^2 = 20^2-12^2 = 34^2-30^2 = 65^2-63^2. - Alois P. Heinz, Aug 06 2019
Number of ways to write 2n as the sum of two positive integers r and s such that r < s and (s - r) | (s * r). - Wesley Ivan Hurt, Apr 21 2020

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

Antti Karttunen, Table of n, a(n) for n = 1..65537
Hanz Becker, Pythagorean triples in JavaScript.
Ron Knott, Pythagorean Triples and Online Calculators.
Project Euler, Problem 176: Right-angled triangles that share a cathetus.
Fred Richman, Pythagorean Triples. [Wayback Machine link]
Amitabha Tripathi, On Pythagorean triples containing a fixed integer, Fib. Q., 46/47 (2008/2009), 331-340. See Theorem 6.
Eric Weisstein's World of Mathematics, Pythagorean Triple.

Cf. A000290, A046080, A046081, A001227, A018892, A024361, A024362, A024363.

Cf. A001620, A082633, A225746.

a[n_] := (DivisorSigma[0, If[OddQ[n], n, n / 2]^2] - 1) / 2; Table[a[i], {i, 100}] (* Amber Hu (hupo001(AT)gmail.com), Jan 23 2008 *)
a[ n_] := Length @ FindInstance[ n > 0 && y > 0 && z > 0 && n^2 + y^2 == z^2, {y, z}, Integers, 10^9]; (* Michael Somos, Jul 25 2018 *)

A046079(n) = ((numdiv(if(n%2, n, n/2)^2)-1)/2); \\ Antti Karttunen, Sep 27 2018

from math import prod
from sympy import factorint
def A046079(n): return prod((e+(p&1)<<1)-1 for p,e in factorint(n).items())>>1 # Chai Wah Wu, Sep 06 2022

def A046079(n) : return (number_of_divisors(n^2 if n%2==1 else n^2/4) - 1) // 2 # Eric M. Schmidt, Jan 26 2013

For odd n, a(n) = A018892(n) - 1.
Let n = (2^a0)*(p1^a1)*...*(pk^ak). Then a(n) = [(2*a0 - 1)*(2*a1 + 1)*(2*a2 + 1)*(2*a3 + 1)*...*(2*ak + 1) - 1]/2. Note that if there is no a0 term, i.e., if n is odd, then the first term is simply omitted. - Temple Keller (temple.keller(AT)gmail.com), Jan 05 2008
For odd n, a(n) = (tau(n^2) - 1) / 2; for even n, a(n) = (tau((n / 2)^2) - 1) / 2. - Amber Hu (hupo001(AT)gmail.com), Jan 23 2008
a(n) = Sum_{i=1..n-1} (1 - ceiling(i*(2*n-i)/(2*n-2*i)) + floor(i*(2*n-i)/(2*n-2*i))). - Wesley Ivan Hurt, Apr 21 2020
Sum_{k=1..n} a(k) ~ (n / Pi^2) * (log(n)^2 + c_1 * log(n) + c_2), where c_1 = 2 * (gamma - 1) + 48*log(A) - 4*log(Pi) - 13*log(2)/3 = 3.512088... (gamma = A001620, log(A) = A225746), and c_2 = 6 * gamma^2 - (6 + log(2)) * gamma + 2 - Pi^2/2 + 19*log(2)^2/18 + log(2)/3 - 6*gamma_1 + 8 * (zeta'(2)/zeta(2))^2 + (4 - 12*gamma + 2*log(2)/3) * zeta'(2)/zeta(2) - 4*zeta''(2)/zeta(2) = -4.457877... (gamma_1 = -A082633). - Amiram Eldar, Nov 08 2024

A046109 Number of lattice points (x,y) on the circumference of a circle of radius n with center at (0,0).

Original entry on oeis.org

1, 4, 4, 4, 4, 12, 4, 4, 4, 4, 12, 4, 4, 12, 4, 12, 4, 12, 4, 4, 12, 4, 4, 4, 4, 20, 12, 4, 4, 12, 12, 4, 4, 4, 12, 12, 4, 12, 4, 12, 12, 12, 4, 4, 4, 12, 4, 4, 4, 4, 20, 12, 12, 12, 4, 12, 4, 4, 12, 4, 12, 12, 4, 4, 4, 36, 4, 4, 12, 4, 12, 4, 4, 12, 12, 20, 4, 4, 12, 4, 12, 4, 12, 4, 4, 36

Offset: 0

Eric W. Weisstein

Also number of Gaussian integers x + yi having absolute value n. - Alonso del Arte, Feb 11 2012

The indices of terms not equaling 4 or 12 correspond to A009177, n>0. - Bill McEachen, Aug 14 2025

			a(5) = 12 because the circumference of the circle with radius 5 will pass through the twelve points (5, 0), (4, 3), (3, 4), (0, 5), (-3, 4), (-4, 3), (-5, 0), (-4, -3), (-3, -4), (0, -5), (3, -4) and (4, -3). Alternatively, we can say the twelve Gaussian integers 5, 4 + 3i, ... , 4 - 3i all have absolute value of 5.

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
Michael Gilleland, Some Self-Similar Integer Sequences
Eric Weisstein's World of Mathematics, Circle Lattice Points

Cf. A004018, A046080, A046110, A046111, A046112, A063014, A256452.

Cf. A000328, A051132, A009177.

a046109 n = length [(x,y) | x <- [-n..n], y <- [-n..n], x^2 + y^2 == n^2]
-- Reinhard Zumkeller, Jan 23 2012

N:= 1000: # to get a(0) to a(N)
A:= Array(0..N):
A[0]:= 1:
for x from 1 to N do
  A[x]:= A[x]+4;
  for y from 1 to min(x-1,floor(sqrt(N^2-x^2))) do
     z:= x^2+y^2;
     if issqr(z) then
       t:= sqrt(z);
       A[t]:= A[t]+8;
     fi
  od
od:
seq(A[i],i=0..N); # Robert Israel, May 08 2015

Table[Length[Select[Flatten[Table[r + I i, {r, -n, n}, {i, -n, n}]], Abs[#] == n &]], {n, 0, 49}] (* Alonso del Arte, Feb 11 2012 *)

a(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)) \\ Charles R Greathouse IV, Feb 01 2017

a(n)=if(n==0, return(1)); t=0; for(x=1, n-1, y=n^2-x^2; if(issquare(y), t++)); return(4*t+4) \\ Arkadiusz Wesolowski, Nov 14 2017

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 4*r if n > 0 else 0
# Orson R. L. Peters, Jan 31 2017

a(n) = A000328(n) - A051132(n).
a(n) = 8*A046080(n) + 4 for n > 0.
a(n) = A004018(n^2).
From Jean-Christophe Hervé, Dec 01 2013: (Start)
a(A084647(k)) = 28.
a(A084648(k)) = 36.
a(A084649(k)) = 44. (End)
a(n) = 4 * Product_{i=1..k} (2*e_i + 1) for n > 0, given that p_i^e_i is the i-th factor of n with p_i = 1 mod 4. - Orson R. L. Peters, Jan 31 2017
a(n) = [x^(n^2)] theta_3(x)^2, where theta_3() is the Jacobi theta function. - Ilya Gutkovskiy, Apr 20 2018
From Hugo Pfoertner, Sep 21 2023: (Start)
a(n) = 8*A063014(n) - 4 for n > 0.
a(n) = 4*A256452(n) for n > 0. (End)

A084647 Hypotenuses for which there exist exactly 3 distinct integer triangles.

Original entry on oeis.org

125, 250, 375, 500, 750, 875, 1000, 1125, 1375, 1500, 1750, 2000, 2197, 2250, 2375, 2625, 2750, 2875, 3000, 3375, 3500, 3875, 4000, 4125, 4394, 4500, 4750, 4913, 5250, 5375, 5500, 5750, 5875, 6000, 6125, 6591, 6750, 7000, 7125, 7375, 7750

Offset: 1

Eric W. Weisstein, Jun 01 2003

nonn

Numbers whose square is decomposable in 3 different ways into the sum of two nonzero squares: these are those with exactly one prime divisor of the form 4k+1 with multiplicity three. - Jean-Christophe Hervé, Nov 11 2013

			a(1) = 125 = 5^3, and 125^2 = 100^2 + 75^2 = 117^2 + 44^2 = 120^2 + 35^2. - _Jean-Christophe Hervé_, Nov 11 2013

Ray Chandler, Table of n, a(n) for n = 1..10000 (first 1140 terms from Jean-Christophe Hervé)
Eric Weisstein's World of Mathematics, Pythagorean Triple

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

Cf. A004144 (0), A084645 (1), A084646 (2), 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), A290505 (19), A097103 (22), A097244 (31), A097245 (37), A097282 (40), A097626 (67).

Clear[lst,f,n,i,k] f[n_]:=Module[{i=0,k=0},Do[If[Sqrt[n^2-i^2]==IntegerPart[Sqrt[n^2-i^2]],k++ ],{i,n-1,1,-1}]; k/2]; lst={}; Do[If[f[n]==3,AppendTo[lst,n]],{n,4*5!}]; lst (* Vladimir Joseph Stephan Orlovsky, Aug 12 2009 *)

Terms are obtained by the products A004144(k)*A002144(p)^3 for k, p > 0, ordered by increasing values. - Jean-Christophe Hervé, Nov 12 2013

A084648 Hypotenuses for which there exist exactly 4 distinct integer triangles.

Original entry on oeis.org

65, 85, 130, 145, 170, 185, 195, 205, 221, 255, 260, 265, 290, 305, 340, 365, 370, 377, 390, 410, 435, 442, 445, 455, 481, 485, 493, 505, 510, 520, 530, 533, 545, 555, 565, 580, 585, 595, 610, 615, 625, 629, 663, 680, 685, 689, 697, 715, 730, 740, 745

Offset: 1

Eric W. Weisstein, Jun 01 2003

nonn

Numbers whose square is decomposable in 4 different ways into the sum of two nonzero squares: these are those with exactly 2 distinct prime divisors of the form 4k+1, each with multiplicity one, or with only one prime divisor of this form with multiplicity 4. - Jean-Christophe Hervé, Nov 11 2013

If m is a term, then 2*m and p*m are terms where p is any prime of the form 4k+3. - Ray Chandler, Dec 30 2019

			a(1) = 65 = 5*13, and 65^2 = 52^2 + 39^2 = 56^2 + 33^2 = 60^2 + 25^2 = 63^2 + 16^2. - _Jean-Christophe Hervé_, Nov 11 2013

Ray Chandler, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Pythagorean Triple

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

Cf. A004144 (0), A084645 (1), A084646 (2), A084647 (3), 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), A290505 (19), A097103 (22), A097244 (31), A097245 (37), A097282 (40), A097626 (67).

Clear[lst,f,n,i,k] f[n_]:=Module[{i=0,k=0},Do[If[Sqrt[n^2-i^2]==IntegerPart[Sqrt[n^2-i^2]],k++ ],{i,n-1,1,-1}]; k/2]; lst={}; Do[If[f[n]==4,AppendTo[lst,n]],{n,6!}]; lst (* Vladimir Joseph Stephan Orlovsky, Aug 12 2009 *)

A084649 Hypotenuses for which there exist exactly 5 distinct Pythagorean triangles.

Original entry on oeis.org

3125, 6250, 9375, 12500, 18750, 21875, 25000, 28125, 34375, 37500, 43750, 50000, 56250, 59375, 65625, 68750, 71875, 75000, 84375, 87500, 96875, 100000, 103125, 112500, 118750, 131250, 134375, 137500, 143750, 146875, 150000, 153125

Offset: 1

Eric W. Weisstein, Jun 01 2003

nonn

Numbers whose square is decomposable in 5 different ways into the sum of two nonzero squares: these are those with exactly one prime divisor of the form 4k+1 with multiplicity 5. - Jean-Christophe Hervé, Nov 12 2013

			a(1) = 5^5, a(5) = 6*5^5, a(65) = 13^5. - _Jean-Christophe Hervé_, Nov 12 2013

Ray Chandler, Table of n, a(n) for n = 1..10000 (first 1019 terms from Jean-Christophe Hervé)
Eric Weisstein's World of Mathematics, Pythagorean Triple

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

Cf. A004144 (0), A084645 (1), A084646 (2), A084647 (3), A084648 (4), 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), A290505 (19), A097103 (22), A097244 (31), A097245 (37), A097282 (40), A097626 (67).

Clear[lst,f,n,i,k] f[n_]:=Module[{i=0,k=0},Do[If[Sqrt[n^2-i^2]==IntegerPart[Sqrt[n^2-i^2]],k++ ],{i,n-1,1,-1}]; k/2]; lst={}; Do[If[f[n]==5,AppendTo[lst,n]],{n,3*6!}]; lst (* Vladimir Joseph Stephan Orlovsky, Aug 12 2009 *)

Terms are obtained by the products A004144(k)*A002144(p)^5 for k, p > 0 ordered by increasing values. - Jean-Christophe Hervé, Nov 12 2013

A084646 Hypotenuses for which there exist exactly 2 distinct integer triangles.

Original entry on oeis.org

25, 50, 75, 100, 150, 169, 175, 200, 225, 275, 289, 300, 338, 350, 400, 450, 475, 507, 525, 550, 575, 578, 600, 675, 676, 700, 775, 800, 825, 841, 867, 900, 950, 1014, 1050, 1075, 1100, 1150, 1156, 1175, 1183, 1200, 1225, 1350, 1352, 1369, 1400

Offset: 1

Eric W. Weisstein, Jun 01 2003

nonn

Numbers whose square is decomposable in 2 different ways into the sum of two nonzero squares: these are those with exactly one prime divisor of the form 4k+1 with multiplicity two. - Jean-Christophe Hervé, Nov 11 2013

Ray Chandler, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Pythagorean Triple

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

Cf. A004144 (0), A084645 (1), 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), A290505 (19), A097103 (22), A097244 (31), A097245 (37), A097282 (40), A097626 (67).

Clear[lst,f,n,i,k] f[n_]:=Module[{i=0,k=0},Do[If[Sqrt[n^2-i^2]==IntegerPart[Sqrt[n^2-i^2]],k++ ],{i,n-1,1,-1}]; k/2]; lst={}; Do[If[f[n]==2,AppendTo[lst,n]],{n,4*5!}]; lst (* Vladimir Joseph Stephan Orlovsky, Aug 12 2009 *)

Terms are obtained by the products A004144(k)*A002144(p)^2 for k, p > 0, ordered by increasing values. - Jean-Christophe Hervé, Nov 12 2013

A046080(a(n)) = 2, A046109(a(n)) = 20. - Jean-Christophe Hervé, Dec 01 2013

A046081 Number of integer-sided right triangles with n as a hypotenuse or leg.

Original entry on oeis.org

0, 0, 1, 1, 2, 1, 1, 2, 2, 2, 1, 4, 2, 1, 5, 3, 2, 2, 1, 5, 4, 1, 1, 7, 4, 2, 3, 4, 2, 5, 1, 4, 4, 2, 5, 7, 2, 1, 5, 8, 2, 4, 1, 4, 8, 1, 1, 10, 2, 4, 5, 5, 2, 3, 5, 7, 4, 2, 1, 14, 2, 1, 7, 5, 8, 4, 1, 5, 4, 5, 1, 12, 2, 2, 9, 4, 4, 5, 1, 11, 4, 2, 1, 13, 8, 1, 5, 7, 2, 8, 5, 4, 4, 1, 5, 13, 2, 2, 7

Offset: 1

Eric W. Weisstein

nonn

Pythagorean triples including primitive ones and non-primitive ones. For a certain n, it may be a leg or the hypotenuse in either a primitive Pythagorean triple, or a non-primitive Pythagorean triple, or both. - Rui Lin, Nov 02 2019

			From _Rui Lin_, Nov 02 2019: (Start)
n=25 is the least number which meets all of following cases:
1. 25 is a leg of a primitive Pythagorean triple (25,312,313), so A024361(25)=1;
2. 25 is the hypotenuse of a primitive Pythagorean triple (7,24,25), so A024362(25)=1;
3. 25 is a leg of a non-primitive Pythagorean triple (25,60,65), so A328708(25)=1;
4. 25 is the hypotenuse of a non-primitive Pythagorean triple (15,20,25), so A328712(25)=1;
5. Combination 1. and 3. means A046079(25)=2;
6. Combination 2. and 4. means A046080(25)=2;
7. Combination 1. and 2. means A024363(25)=2;
8. Combination 3. and 4. means A328949(25)=2;
9. Combination of 1., 2., 3., and 4. means A046081(25)=4. (End)

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

Lars Blomberg, Table of n, a(n) for n = 1..10000
Anonymous, Generator of all Pythagorean triples that include a given number [Internet Archive Wayback Machine]
Ron Knott, Pythagorean Triples and Online Calculators
F. Richman, Pythagorean Triples
A. Tripathi, On Pythagorean triples containing a fixed integer, Fib. Q., 46/47 (2008/2009), 331-340. See Theorem 8.
Eric Weisstein's World of Mathematics, Pythagorean Triple

Cf. A006593, A046079, A046080, A024361, A024362, A024363, A328708, A328712, A328949.

a[1] = 0; a[n_] := Module[{f}, f = Select[FactorInteger[n], Mod[#[[1]], 4] == 1&][[All, 2]]; (DivisorSigma[0, If[OddQ[n], n, n/2]^2]-1)/2 + (Times @@ (2*f+1) - 1)/2]; Array[a, 99] (* Jean-François Alcover, Jul 19 2017 *)

a(n) = {oddn = n/(2^valuation(n, 2)); f = factor(oddn); for (k=1, #f~, if ((f[k,1] % 4) != 1, f[k,2] = 0);); n1 = factorback(f); if (n % 2, (numdiv(n^2)+numdiv(n1^2))/2 -1, (numdiv((n/2)^2)+numdiv(n1^2))/2 -1);} \\ Michel Marcus, Mar 07 2016

from sympy import factorint
def a(n):
    p1, p2 = 1, 1
    for i in factorint(n).items():
        if i[0] % 4 == 1:
            p2 *= i[1] * 2 + 1
        p1 *= i[1] * 2 + 1 - (2 if i[0] == 2 else 0)
    return (p1 + p2)//2 - 1
print([a(n) for n in range(1, 100)])  # Oleg Sorokin, Mar 02 2023

a(n) = A046079(n) + A046080(n). - Lekraj Beedassy, Dec 01 2003
From Rui Lin, Nov 02 2019: (Start)
a(n) = A024363(n) + A328949(n).
a(n) = A024361(n) + A024362(n) + A328708(n) + A328712(n). (End)

Improved name by Bernard Schott, Jan 03 2019

cp's OEIS Frontend

A097756 Table read by rows of A054994 ordered by A046080.

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A083025 Number of primes congruent to 1 modulo 4 dividing n (with multiplicity).

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A084645 Hypotenuses for which there exists a unique integer-sided right triangle.

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A046079 Number of Pythagorean triangles with leg n.

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A046109 Number of lattice points (x,y) on the circumference of a circle of radius n with center at (0,0).

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A084647 Hypotenuses for which there exist exactly 3 distinct integer triangles.

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A084648 Hypotenuses for which there exist exactly 4 distinct integer triangles.

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