A224921 -id:A224921 - cp's OEIS frontend

A046080 a(n) is the number of integer-sided right triangles with hypotenuse n.

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, 4, 0, 0, 1, 0, 1, 0, 0, 1, 1, 2, 0, 0, 1, 0, 1, 0, 1, 0, 0, 4, 0, 1, 0, 1, 1, 1, 0, 0, 0, 1, 0, 1, 0, 0

Offset: 1

Eric W. Weisstein

nonn

Or number of ways n^2 can be written as the sum of two positive squares: a(5) = 1: 3^2 + 4^2 = 5^2; a(25) = 2: 7^2 + 24^2 = 15^2 + 20^2 = 25^2. - Alois P. Heinz, Aug 01 2019

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

Stanislav Sykora, Table of n, a(n) for n = 1..20000
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 7.
Eric Weisstein's World of Mathematics, Pythagorean Triple

First differs from A083025 at n=65.
Cf. A000290, A006339, A024362, A046079, A046081, A009000.
A088111 gives records; A088959 gives where records occur.
Cf. A046109, A063725.
Cf. A004144, A084647, A084648, A084649.
Partial sums: A224921.

f:= proc(n) local F,t;
  F:= select(t -> t[1] mod 4 = 1, ifactors(n)[2]);
  1/2*(mul(2*t[2]+1, t=F)-1)
end proc:
map(f, [$1..100]); # Robert Israel, Jul 18 2016

a[1] = 0; a[n_] := With[{fi = Select[ FactorInteger[n], Mod[#[[1]], 4] == 1 & ][[All, 2]]}, (Times @@ (2*fi+1)-1)/2]; Table[a[n], {n, 1, 99}] (* Jean-François Alcover, Feb 06 2012, after first formula *)

a(n)={my(m=0,k=n,n2=n*n,k2,l2);
while(1,k=k-1;k2=k*k;l2=n2-k2;if(l2>k2,break);if(issquare(l2),m++));return(m)} \\ brute force, Stanislav Sykora, Mar 18 2015

{a(n) = if( n<1, 0, sum(k=1, sqrtint(n^2 \ 2), issquare(n^2 - k^2)))}; /* Michael Somos, Mar 29 2015 */

a(n) = {my(f = factor(n/(2^valuation(n, 2)))); (prod(k=1, #f~, if ((f[k,1] % 4) == 1, 2*f[k,2] + 1, 1)) - 1)/2;} \\ Michel Marcus, Mar 08 2016

from math import prod
from sympy import factorint
def A046080(n): return prod((e<<1)+1 for p,e in factorint(n).items() if p&3==1)>>1 # Chai Wah Wu, Sep 06 2022

Let n = 2^e_2 * product_i p_i^f_i * product_j q_j^g_j where p_i == 1 mod 4, q_j == 3 mod 4; then a(n) = (1/2)*(product_i (2*f_i + 1) - 1). - Beiler, corrected
8*a(n) + 4 = A046109(n) for n > 0. - Ralf Stephan, Mar 14 2004
a(n) = 0 for n in A004144. - Lekraj Beedassy, May 14 2004
a(A084645(k)) = 1. - Ruediger Jehn, Jan 14 2022
a(A084646(k)) = 2. - Ruediger Jehn, Jan 14 2022
a(A084647(k)) = 3. - Jean-Christophe Hervé, Dec 01 2013
a(A084648(k)) = 4. - Jean-Christophe Hervé, Dec 01 2013
a(A084649(k)) = 5. - Jean-Christophe Hervé, Dec 01 2013
a(n) = A063725(n^2) / 2. - Michael Somos, Mar 29 2015
a(n) = Sum_{k=1..n} Sum_{i=1..k} [i^2 + k^2 = n^2], where [ ] is the Iverson bracket. - Wesley Ivan Hurt, Dec 10 2021
a(A002144(k)^n) = n. - Ruediger Jehn, Jan 14 2022

A247588 Number of integer-sided acute triangles with largest side n.

Original entry on oeis.org

1, 2, 3, 5, 6, 8, 11, 13, 15, 17, 21, 25, 27, 31, 34, 39, 43, 48, 52, 56, 63, 67, 73, 80, 84, 90, 96, 104, 111, 117, 126, 132, 140, 147, 154, 165, 172, 183, 189, 198, 210, 219, 229, 237, 247, 260, 270, 282, 292, 302

Offset: 1

Vladimir Letsko, Sep 20 2014

nonn

			a(3) = 3 because there are 3 integer-sided acute triangles with largest side 3: (1,3,3); (2,3,3); (3,3,3).

Vladimir Letsko, Mathematical Marathon, problem 192 (in Russian).

Cf. A046080, A002623, A224921, A247586, A247587, A247589.

tr_a:=proc(n) local a,b,t,d;t:=0:
for a to n do
for b from max(a,n+1-a) to n do
d:=a^2+b^2-n^2:
if d>0 then t:=t+1 fi
od od;
t; end;

a[ n_] := Length @ FindInstance[ n >= b >= a >= 1 && n < b + a && n^2 < b^2 + a^2, {a, b}, Integers, 10^9]; (* Michael Somos, May 24 2015 *)

a(n) = sum(j=0, n*(1 - sqrt(2)/2), n - j - floor(sqrt(2*j*n - j^2))); \\ Michel Marcus, Oct 07 2014

{a(n) = sum(j=0, n - sqrtint(n*n\2) - 1, n - j - sqrtint(2*j*n - j*j))}; /* Michael Somos, May 24 2015 */

a(n) = Sum_{j=0..floor(n*(1 - sqrt(2)/2))} (n - j - floor(sqrt(2*j*n - j^2))). - Anton Nikonov, Oct 06 2014

a(n) = (1/8)*(-4*ceiling((n - 1)/sqrt(2)) + 4*n^2 - A000328(n) + 1), n > 1. - Mats Granvik, May 23 2015

A379830 a(n) is the number of Pythagorean triples (u, v, w) for which w - u = n where u < v < w.

Original entry on oeis.org

0, 0, 1, 0, 1, 0, 1, 0, 2, 2, 1, 0, 1, 0, 1, 0, 2, 0, 4, 0, 1, 0, 1, 0, 2, 3, 1, 2, 1, 0, 1, 0, 5, 0, 1, 0, 4, 0, 1, 0, 2, 0, 1, 0, 1, 2, 1, 0, 2, 4, 7, 0, 1, 0, 4, 0, 2, 0, 1, 0, 1, 0, 1, 2, 5, 0, 1, 0, 1, 0, 1, 0, 8, 0, 1, 3, 1, 0, 1, 0, 2, 6, 1, 0, 1, 0, 1, 0

Offset: 0

Felix Huber, Jan 07 2025

nonn

The difference between the hypotenuse and the short leg of a primitive Pythagorean triple (p^2 - q^2, 2*p*q, p^2 + q^2) (where p > q are coprimes and not both odd) is d = max(2*q^2, (p - q)^2). For every of these primitive Pythagorean triples whose d divides n, there is a Pythagorean triple with w - u = n. Therefore d <= n and it follows that 1 <= q <= sqrt(n/2) and q + 1 <= p <= q + sqrt(n), which means that there is a finite number of Pythagorean triples with w - u = n.

			The a(18) = 4 Pythagorean triples are (27, 36, 45), (16, 30, 34), (40, 42, 58), (7, 24, 25) because 45 - 27 = 34 - 16 = 58 - 40 = 25 - 7 = 18.
See also linked Maple program "Pythagorean triples for which w - u = n".

Felix Huber, Table of n, a(n) for n = 0..10000
Felix Huber, Pythagorean triples for which w - u = n
Eric Weisstein's World of Mathematics, Pythagorean Triple

Cf. A024361, A024362, A024363, A046079, A046080, A046081, A096033, A224921.

A379830:=proc(n)
    local a,p,q;
    a:=0;
    for q to isqrt(floor(n/2)) do
        for p from q+1 to q+isqrt(n) do
            if igcd(p,q)=1 and (is(p,even) or is(q,even)) and n mod max((p-q)^2,2*q^2)=0 then
                a:=a+1
            fi
        od
    od;
    return a
end proc;
seq(A379830(n),n=0..87);

A247587 Number of obtuse triangles with integer sides at most n.

Original entry on oeis.org

0, 0, 1, 2, 4, 8, 13, 20, 30, 42, 57, 74, 95, 120, 149, 182, 219, 261, 309, 362, 420, 485, 556, 632, 715, 806, 906, 1012, 1125, 1247, 1377, 1517, 1666, 1824, 1993, 2170, 2358, 2555, 2765, 2986

Offset: 1

Vladimir Letsko, Sep 20 2014

nonn

			a(4) = 2 because there are 2 obtuse triangles with integer sides less than or equal to 4: (2,2,3); (2,3,4).

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Vladimir Letsko, Mathematical Marathon, problem 192 (in Russian).

Cf. A002623, A224921, A247586, A247588, A247589.

tr_o:=proc(n) local a, b, c, t, d; t:=0:
  for a to n do
  for b from a to n do
  for c from b to min(a+b-1, n) do
  d:=a^2+b^2-c^2:
  if d<0 then t:=t+1 fi
  od od od;
  [n, t]; end;

a(n)=sum(a=2,n-1,sum(b=a,n-1,max(0,min(n,a+b-1)-sqrtint(a^2+b^2)))) \\ Charles R Greathouse IV, Sep 20 2014

obtuse(n)=sum(a=2,n-1, max(0, sqrtint(n^2-1-a^2)-max(a,n-a+1)+1))
s=0; vector(100,n, s+=obtuse(n)) \\ Charles R Greathouse IV, Sep 20 2014

A156685 Number of primitive Pythagorean triples A^2 + B^2 = C^2 with 0 < A < B < C and gcd(A,B)=1 that have a hypotenuse C that is less than or equal to n.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 11, 11, 11, 11, 11, 11, 11, 11, 12, 12, 12

Offset: 1

Ant King, Feb 17 2009

D. N. Lehmer has proved that the asymptotic density of a(n) is a(n)/n = 1/(2*Pi) = 0.1591549...

			There is one primitive Pythagorean triple with a hypotenuse less than or equal to 7 -- (3,4,5) -- hence a(7)=1.
G.f. = x^5 + x^6 + x^7 + x^8 + x^9 + x^10 + x^11 + x^12 + 2*x^13 + 2*x^14 + ...

Lehmer, Derrick Norman; Asymptotic Evaluation of Certain Totient Sums, American Journal of Mathematics, Vol. 22, No. 4, (Oct. 1900), pp. 293-335.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Ron Knott, Right-angled Triangles and Pythagoras' Theorem
Ramin Takloo-Bighash, How many Pythagorean triples are there?, A Pythagorean Introduction to Number Theory, Undergraduate Texts in Mathematics, Springer, Cham, 2018, 211-226.

Cf. A008846, A020882, A024409, A024362, A224921.

a156685 n = a156685_list !! (n-1)
a156685_list = scanl1 (+) a024362_list  -- Reinhard Zumkeller, Dec 02 2012

RightTrianglePrimitiveHypotenuses[1]:=0;RightTrianglePrimitiveHypotenuses[n_Integer?Positive]:=Module[{f=Transpose[FactorInteger[n]],a,p,mod1posn},{p,a}=f;mod1=Select[p,Mod[ #,4]==1&];If[Length[a]>Length[mod1],0,2^(Length[mod1]-1)]];RightTrianglePrimitiveHypotenuses[ # ] &/@Range[75]//Accumulate

a(n)=sum(a=1,n-2,sum(b=a+1,sqrtint(n^2-a^2), gcd(a,b)==1 && issquare(a^2+b^2))) \\ Charles R Greathouse IV, Apr 29 2013

Essentially partial sums of A024362.

A247586 Number of acute triangles with integer sides less than or equal to n.

Original entry on oeis.org

1, 3, 6, 11, 17, 25, 36, 49, 64, 81, 102, 127, 154, 185, 219, 258, 301, 349, 401, 457, 520, 587, 660, 740, 824, 914, 1010, 1114, 1225, 1342, 1468, 1600, 1740, 1887, 2041, 2206, 2378, 2561, 2750, 2948

Offset: 1

Vladimir Letsko, Sep 20 2014

nonn

			a(2) = 3 because there are 3 acute triangles with integer sides less than or equal to 2: (1,1,1); (1,2,2); (2,2,2).

Vladimir Letsko, Mathematical Marathon, problem 192 (in Russian).

Cf. A002623, A224921, A247587, A247588, A247589.

tr_a:=proc(n) local a,b,c,t,d;t:=0:
  for a to n do
  for b from a to n do
  for c from b to min(a+b-1,n) do
  d:=a^2+b^2-c^2:
  if d>0 then t:=t+1 fi
  od od od;
  [n,t]; end;

a[n_] := Module[{a, b, c, d, t = 0}, Do[d = a^2 + b^2 - c^2; If[d>0, t++], {a, n}, {b, a, n}, {c, b, Min[a+b-1, n]}]; t]; Array[a, 40] (* Jean-François Alcover, Jun 19 2019, from Maple *)

import itertools
def A247586(n):
    I = itertools.combinations_with_replacement(range(1,n+1),3)
    F = filter(lambda c: c[0]**2 + c[1]**2 > c[2]**2, I)
    return len(list(F))
print([A247586(n) for n in range(41)]) # Peter Luschny, Sep 22 2014

A224932 Maximal side length b <= a = n of integer parallelograms.

Original entry on oeis.org

0, 0, 0, 3, 5, 0, 6, 6, 8, 10, 10, 11, 13, 13, 15, 15, 17, 16, 18, 20, 20, 21, 21, 23, 25, 26, 26, 27, 29, 30, 29, 31, 31, 34, 35, 33, 37, 37, 39, 40, 41, 41, 41, 43, 45, 45, 46, 46, 48, 50

Offset: 1

Reiner Moewald, Apr 20 2013

nonn

Reiner Moewald, Table of n, a(n) for n = 1..100

Cf. A224921, A224923.

A247589 Number of integer-sided obtuse triangles with largest side n.

Original entry on oeis.org

0, 0, 1, 1, 2, 4, 5, 7, 10, 12, 15, 17, 21, 25, 29, 33, 37, 42, 48, 53, 58, 65, 71, 76, 83, 91, 100, 106, 113, 122, 130, 140, 149, 158, 169, 177, 188, 197, 210, 221, 230, 243, 255, 269, 281, 292, 306, 318, 333, 346

Offset: 1

Vladimir Letsko, Sep 20 2014

nonn

			a(5) = 2 because there are 2 integer-sided acute triangles with largest side 5: (2,4,5); (3,3,5).

Vladimir Letsko, Mathematical Marathon, problem 192 (in Russian).

Cf. A046080, A002623, A224921, A247586, A247587, A247588.

tr_o:=proc(n) local a,b,t,d;t:=0:
for a to n do
for b from max(a,n+1-a) to n do
d:=a^2+b^2-n^2:
if d<0 then t:=t+1 fi
od od;
t; end;

a(n) = k*(k + (1+(-1)^n)/2) + Sum_{j=1..floor(n*(1-sqrt(2)/2))} floor(sqrt(2*j*n - j^2 - 1) - j), where k = floor((2*n*(sqrt(2) - 1) + 1 - (-1)^n)/4) (it appears that k(n) is A070098(n)). - Anton Nikonov, Sep 29 2014

A383181 Family of 2-colorings of {1..7824} with no monochromatic Pythagorean triples.

Original entry on oeis.org

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

Offset: 1

David Dewan, Apr 18 2025

We use the codes: 1=red, 2=blue, 0=unconstrained (may be red or blue).
Choose any Pythagorean triangle (r,s,t) with t<=7824, then a(r), a(s), a(t) cannot all be the same color (see Examples).
Solution and proof by Heule, Kullmann, and Marek (2016).
Because each of the 2899 numbers for which a(n)=0 can be independently colored red or blue, this sequence represents 2^2899 unique 2-colorings with no monochromatic Pythagorean triples.
There is no 2-coloring of {1..7825} with no monochromatic Pythagorean triples.

			The triple (5,12,13) is not monochromatic:
  a(5)= 1 red,
  a(12)=2 blue,
  a(13)=2 blue.
The triple (3,4,5) is not monochromatic whether 4 is red or blue:
  a(3)=2 blue,
  a(4)=0 red or blue,
  a(5)=1 red.

David Dewan, Table of n, a(n) for n = 1..7824
David Dewan, Image Decoder Program for Boolean Pythagorean Triples
Marijn Heule, Visualization of a solution of the Pythagorean Triples Problem, 21 May 2016.
Marijn J. H. Heule, Oliver Kullmann, and Victor W. Marek, Solving and Verifying the boolean Pythagorean Triples problem via Cube-and-Conquer, arXiv:1605.00723 [cs.DM], 3 May 2016.
Wikipedia, Boolean Pythagorean triples problem.
Bob Yirka, Computer generated math proof is largest ever at 200 terabytes, Phys.org, 30 May 2016.

Cf. A272709, A224921, A156685, A009003.

cp's OEIS Frontend

A046080 a(n) is the number of integer-sided right triangles with hypotenuse n.

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A247588 Number of integer-sided acute triangles with largest side n.

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A379830 a(n) is the number of Pythagorean triples (u, v, w) for which w - u = n where u < v < w.

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A247587 Number of obtuse triangles with integer sides at most n.

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A156685 Number of primitive Pythagorean triples A^2 + B^2 = C^2 with 0 < A < B < C and gcd(A,B)=1 that have a hypotenuse C that is less than or equal to n.

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A247586 Number of acute triangles with integer sides less than or equal to n.

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A224932 Maximal side length b <= a = n of integer parallelograms.

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