A009012 -id:A009012 - cp's OEIS frontend

A009000 Ordered hypotenuse numbers (squares are sums of 2 distinct nonzero squares).

5, 10, 13, 15, 17, 20, 25, 25, 26, 29, 30, 34, 35, 37, 39, 40, 41, 45, 50, 50, 51, 52, 53, 55, 58, 60, 61, 65, 65, 65, 65, 68, 70, 73, 74, 75, 75, 78, 80, 82, 85, 85, 85, 85, 87, 89, 90, 91, 95, 97, 100, 100, 101, 102, 104, 105, 106, 109, 110, 111, 113, 115, 116, 117, 119, 120

Offset: 1

David W. Wilson

The largest member 'c' of the Pythagorean triples (a,b,c) ordered by increasing c.
If c^2 = a^2 + b^2 (a < b < c) then c^2 = (n^2 + m^2)/2 with n = b - a, m = b + a. - Zak Seidov, Mar 03 2011
Numbers n such that A083025(n) > 0, i.e., n is divisible by at least one prime of the form 4k+1. - Max Alekseyev, Oct 24 2008
A number appears only once in the sequence if and only if it is divisible by exactly one prime of the form 4k+1 with multiplicity one (cf. A084645). - Jean-Christophe Hervé, Nov 11 2013
If c^2 = a^2 + b^2 with a and b > 0, then a <> b: the sum of 2 equal squares cannot be a square because sqrt(2) is not rational. - Jean-Christophe Hervé, Nov 11 2013

W. L. Schaaf, Recreational Mathematics, A Guide To The Literature, "The Pythagorean Relationship", Chapter 6 pp. 89-99 NCTM VA 1963.
W. L. Schaaf, A Bibliography of Recreational Mathematics, Vol. 2, "The Pythagorean Relation", Chapter 6 pp. 108-113 NCTM VA 1972.
W. L. Schaaf, A Bibliography of Recreational Mathematics, Vol. 3, "Pythagorean Recreations", Chapter 6 pp. 62-6 NCTM VA 1973.

Zak Seidov and T. D. Noe, Table of n, a(n) for n = 1..10000 (Zak Seidov entered the first 1981 terms).
Anonymous, Links to Pythagorean Theorem Proofs
Henry Bottomley, Pythagoras's theorem (animated proof)
Kangourou Math Website, L'animation du theoreme de Pythagore
Ron Knott, Pythagorean Triples and Online Calculators
Ron Knott, Right-angled Triangles and Pythagoras' Theorem
I. Kobayashi et al., Pythagorean Theorem (Java Interactive Proofs, Applications and Explorations)
Mathematical Database, Poster, 7 Ways to prove the Pythagorean Theorem
B. Richmond, The Pythagorean Theorem
M. Shepperd, Web Resources on Pythagoras' Theorem
J. S. Silverman, A Friendly Introduction to Number Theory, Chapters 1 to 6 (see Chapters 2 and 3).
G. Villemin's Almanach of Numbers, Triangles & Triplets de Pythagore
Eric Weisstein's World of Mathematics, Pythagorean Triple
Index entries for sequences related to sums of squares

Cf. A009012, A009003, A024507, A004431, A046083, A046084, A004144, A083025, A084645, A084646, A084647, A084648, A084649, A006339.

A009000:=proc(N) # To get all terms <= N
    local p,q,i,L;
    L:=[];
    for p from 2 to floor(sqrt(N-1)) do
        for q to p-1 do
            if igcd(p,q)=1 and is(p-q,odd) then
                L:=[op(L),seq(i*(p^2+q^2),i=1..N/(p^2+q^2))];
            fi
        od
    od;
    return op(sort(L))
end proc:
A009000(120); # Felix Huber, Feb 10 2025

max = 120; hypotenuseQ[n_] := For[k = 1, True, k++, p = Prime[k]; Which[Mod[p, 4] == 1 && Divisible[n, p], Return[True], p > n, Return[False]]]; hypotenuses = Select[Range[max], hypotenuseQ]; red[c_] := {a, b, c} /. {ToRules[ Reduce[0 < a <= b && a^2 + b^2 == c^2, {a, b}, Integers]]}; A009000 = Flatten[red /@ hypotenuses, 1][[All, -1]] (* Jean-François Alcover, May 23 2012, after Max Alekseyev *)
Sqrt[#]&/@Flatten[Table[Total/@Select[IntegerPartitions[n^2,{2}],Length[Union[#]]==2&&AllTrue[Sqrt[#],IntegerQ]&],{n,150}]] (* Harvey P. Dale, May 25 2025 *)

list(lim)=my(v=List(),m2,s2,h2,h); for(middle=4,lim-1, m2=middle^2; for(small=1,middle, s2=small^2; if(issquare(h2=m2+s2,&h), if(h>lim, break); listput(v, h)))); vecsort(Vec(v)) \\ Charles R Greathouse IV, Jun 23 2017

list(lim) = {my(lh = List()); for(u = 2, sqrtint(lim), for(v = 1, u, if (u^2+v^2 > lim, break); if ((gcd(u,v) == 1) && (0 != (u-v)%2), for (i = 1, lim, if (i*(u^2+v^2) > lim, break); /* if (u^2 - v^2 < 2*u*v, w = [i*(u^2 - v^2), i*2*u*v, i*(u^2+v^2)], w = [i*2*u*v, i*(u^2 - v^2), i*(u^2+v^2)]); */ listput(lh, i*(u^2+v^2)););););); vecsort(Vec(lh));} \\ Michel Marcus, Apr 10 2021

from math import isqrt
def aupto(limit):
  s = [i*i for i in range(1, limit+1)]
  s2 = sorted(a+b for i, a in enumerate(s) for b in s[i+1:])
  return [isqrt(k) for k in s2 if k in s]
print(aupto(120)) # Michael S. Branicky, May 10 2021

A009023 Long legs of Pythagorean triangles.

Original entry on oeis.org

4, 8, 12, 15, 16, 20, 21, 24, 28, 30, 32, 35, 36, 40, 42, 44, 45, 48, 52, 55, 56, 60, 63, 64, 68, 70, 72, 75, 76, 77, 80, 84, 88, 90, 91, 92, 96, 99, 100, 104, 105, 108, 110, 112, 116, 117, 120, 124, 126, 128, 132, 135, 136, 140, 143, 144, 147, 148, 150, 152, 153, 154, 156

Offset: 1

David W. Wilson

nonn

A227481(a(n)) > 1. - Reinhard Zumkeller, Oct 11 2013
This is A009012 (sorted A046084) without duplicates. - Andrey Zabolotskiy, Dec 27 2017
Does a(n)/n converge to some limit? - Benoit Cloitre, Oct 18 2009
For n = {52000, 72000, 100000}, n/a(n) = {0.499, 0.50175, 0.50428}. - Alex Ratushnyak, Jan 17 2019

Wacław Sierpiński, Pythagorean triangles, Dover books. [Benoit Cloitre, Oct 17 2009]

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Cf. A074235 (complement), A009012, A046084, A227481.

a009023 n = a009023_list !! (n-1)
a009023_list = filter ((> 1) . a227481) [1..]
-- Reinhard Zumkeller, Oct 11 2013

A057096 Saint-Exupéry numbers: ordered products of the three sides of Pythagorean triangles.

Original entry on oeis.org

60, 480, 780, 1620, 2040, 3840, 4200, 6240, 7500, 12180, 12960, 14760, 15540, 16320, 20580, 21060, 30720, 33600, 40260, 43740, 49920, 55080, 60000, 65520, 66780, 79860, 92820, 97440, 97500, 103680, 113400, 118080, 120120, 124320, 130560, 131820, 164640

Offset: 1

Henry Bottomley, Aug 01 2000

nonn

It is an open question whether any two distinct Pythagorean Triples can have the same product of their sides.
From Amiram Eldar, Nov 22 2020: (Start)
Named after the French writer Antoine de Saint-Exupéry (1900-1944).
The problem of finding two distinct Pythagorean triples with the same product was proposed by Eckert (1984). It is equivalent of finding a nontrivial solution of the Diophantine equation x*y*(x^4-y^4) = z*w*(z^4-w^4) (Bremner and Guy, 1988). (End)

			a(1) = 3*4*5 = 60.

Richard K. Guy, "Triangles with Integer Sides, Medians and Area." D21 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 188-190, 1994.
Antoine de Saint-Exupéry, Problème du Pharaon, Liège : Editions Dynamo, 1957.

T. D. Noe, Table of n, a(n) for n = 1..10000
Andrew Bremner and Richard K. Guy, A Dozen Difficult Diophantine Dilemmas, The American Mathematical Monthly, Vol. 95, No. 1 (1988), pp. 31-36.
Ernest J. Eckert, Problem 994, Crux Mathematicorum, Vol. 10, No. 10 (1984), p. 318, entire issue.
Richard K. Guy, Comment to Problem 994, Crux Mathematicorum, Vol. 12, No. 5 (1986), p. 109, entire issue.
Henry Plane, Calcule-moi un parallélépipède..., AMPEP, PLOT No. 22 (2002), pp. 22-23.
Giovanni Resta, Saint-Exupery numbers.
Antoine de Saint Exupéry, Le Problème du Pharaon, Succession Saint Exupéry - d'Agay, 2018.
Eric Weisstein's World of Mathematics, Pythagorean Triple.

Cf. A009004, A009012, A009111, A020886, A057097, A057098, A057099, A057100.

k=5000000; lst={}; Do[Do[If[IntegerQ[a=Sqrt[c^2-b^2]], If[a>=b, Break[]]; x=a*b*c; If[x<=k, AppendTo[lst,x]]], {b,c-1,4,-1}], {c,5,400,1}]; Union@lst (* Vladimir Joseph Stephan Orlovsky, Sep 05 2009 *)

a(n) = 60*A057097(n) = A057098(n)*A057099(n)*A057100(n).

A156681 Consider all Pythagorean triangles A^2 + B^2 = C^2 with A < B < C; sequence gives values of B, sorted to correspond to increasing A (A009004).

Original entry on oeis.org

4, 12, 8, 24, 15, 12, 40, 24, 60, 16, 35, 84, 48, 20, 36, 112, 30, 63, 144, 24, 80, 180, 21, 48, 99, 28, 72, 220, 120, 264, 32, 45, 70, 143, 60, 312, 168, 36, 120, 364, 45, 96, 195, 420, 40, 72, 224, 480, 60, 126, 255, 44, 56, 180, 544, 288, 84, 120, 612, 48, 77, 105

Offset: 1

Ant King, Feb 17 2009

The ordered sequence of B values is A009012(n) (allowing repetitions) and A009023(n) (excluding repetitions).

			As the first four Pythagorean triples (ordered by increasing A) are (3,4,5), (5,12,13), (6,8,10) and (7,24,25), then a(1)=4, a(2)=12, a(3)=8 and a(4)=24.

Albert H. Beiler, Recreations In The Theory Of Numbers, Chapter XIV, The Eternal Triangle, Dover Publications Inc., New York, 1964, pp. 104-134.
W. Sierpinski, Pythagorean Triangles, Dover Publications, Inc., Mineola, New York, 2003.

Shujing Lyu, Table of n, a(n) for n = 1..5000
Ron Knott, Right-angled Triangles and Pythagoras' Theorem
Robert Recorde, The Whetstone of Witte, whiche is the seconde parte of Arithmeteke: containing the extraction of rootes; the cossike practise, with the rule of equation; and the workes of Surde Nombers, London, 1557. See p. 57.

Cf. A156682, A009004, A009012, A009023.

PythagoreanTriplets[n_]:=Module[{t={{3,4,5}},i=4,j=5},While[i

a(n) = sqrt(A156682(n)^2 - A009004(n)^2).

A057099 Middle side of a Pythagorean triangle (ordered by the product of the sides).

Original entry on oeis.org

4, 8, 12, 12, 15, 16, 24, 24, 20, 21, 24, 40, 35, 30, 28, 36, 32, 48, 60, 36, 48, 45, 40, 63, 45, 44, 84, 42, 60, 48, 72, 80, 56, 70, 60, 52, 56, 72, 112, 55, 99, 60, 77, 64, 75, 84, 96, 80, 68, 120, 63, 72, 144, 120, 96, 76, 105, 90, 72, 80, 143, 126, 120, 90, 84, 108, 91

Offset: 1

Henry Bottomley, Aug 01 2000

nonn

			a(1)=4 since 3*4*5=60 is smallest possible positive product

Cf. A009012, A009023, A046084, A057096.

maxShortLeg = 66; terms = 67;
r[a_] := {a, b, c} /. {ToRules[Reduce[a <= b < c && a^2+b^2 == c^2, {b, c}, Integers]]};
abc = r /@ Complement[Range[maxShortLeg], {1, 2, 4}] // Flatten[#, 1]&;
SortBy[abc, Times @@ # &][[;; terms, 2]] (* Jean-François Alcover, Nov 21 2019 *)

a(n) =A057096(n)/(A057098(n)*A057100(n)) =sqrt(A057100(n)^2-A057098(n)^2)

A074235 Numbers that cannot be a long leg of an integer right triangle.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 9, 10, 11, 13, 14, 17, 18, 19, 22, 23, 25, 26, 27, 29, 31, 33, 34, 37, 38, 39, 41, 43, 46, 47, 49, 50, 51, 53, 54, 57, 58, 59, 61, 62, 65, 66, 67, 69, 71, 73, 74, 78, 79, 81, 82, 83, 85, 86, 87, 89, 93, 94, 95, 97, 98, 101, 102, 103, 106, 107, 109, 111

Offset: 1

Zak Seidov, Sep 18 2002

nonn

The possible values of a long leg of an integer right triangle are in A009012.

A227481(a(n)) = 1. - Reinhard Zumkeller, Oct 11 2013

			5 is a term because a^2 + 5^2 = c^2 has no solution for a < 5 with integers a, c.
13 is a term because a^2 + 13^2 = c^2 has no solution for a < 13 with integers a, c.

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Cf. A009012.

Cf. A009023 (complement).

a074235 n = a074235_list !! (n-1)
a074235_list = filter ((== 1) . a227481) [1..]
-- Reinhard Zumkeller, Oct 11 2013

A376900 a(n) is the number of distinct integer-sided right triangles that can be drawn into a square with side length n.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 1, 1, 2, 2, 2, 3, 4, 4, 5, 5, 6, 6, 6, 6, 8, 10, 10, 10, 11, 11, 11, 11, 13, 13, 14, 14, 17, 17, 17, 18, 18, 18, 18, 19, 20, 21, 23, 25, 25, 25, 25, 27, 27, 27, 27, 29, 31, 31, 31, 34, 34, 34, 34, 37, 37, 37, 37, 40, 41, 42, 43, 43, 45, 45, 46

Offset: 0

Felix Huber, Oct 25 2024

nonn

			a(11) = 3 because exactly the 3 integer-sided right triangles (3, 4, 5), (6, 8, 10), (5, 12, 13) can be drawn into a square with side length 11.
See linked Maple program to calculate the right triangles for a given n.

Felix Huber, Table of n, a(n) for n = 0..10000
Felix Huber, Maple program to calculate the right triangles for a given n

Cf. A009012.

A376900:=proc(n)
   local a,p,q,v,k;
   a:=0;
      for p from 2 to evalf(sqrt(sqrt(2)*n+1)) do
         for q from 1 to min(p-1,floor(n/(sqrt(2)*p))) do
            if gcd(p,q)=1 and is(p+q,odd) then
               v:=max(p^2-q^2,2*p*q);
               k:=min(p^2-q^2,2*p*q)/v;
               a:=a+floor(n/v*sqrt(k^2-2*k+2));
            fi;
         od;
      od;
   return a;
end proc;
seq(A376900(n),n=0..70);

A009041 Ordered legs of Pythagorean triangles.

Original entry on oeis.org

3, 4, 5, 6, 7, 8, 8, 9, 9, 10, 11, 12, 12, 12, 12, 13, 14, 15, 15, 15, 15, 16, 16, 16, 17, 18, 18, 19, 20, 20, 20, 20, 21, 21, 21, 21, 22, 23, 24, 24, 24, 24, 24, 24, 24, 25, 25, 26, 27, 27, 27, 28, 28, 28, 28, 29, 30, 30, 30, 30, 31, 32, 32, 32, 32, 33, 33, 33, 33, 34, 35, 35, 35, 35, 36

Offset: 1

David W. Wilson

nonn

Order the set of all Pythagorean triangles. This is the sequence of the first leg.

Sorted union of A009004 and A009012 retaining duplicates. - Sean A. Irvine, Apr 18 2018

			First legs of (3,4,5), (4,3,5), (5,12,13), (6,8,10), (7,24,25), (8,6,10), (8,15,17), (9,12,15), (9,40,41), ... - _Michael Somos_, Mar 03 2004

Ron Knott, Pythagorean Triples and Online Calculators

Cf. A009004, A009012.

A009070 Ordered sides of Pythagorean triangles.

Original entry on oeis.org

3, 4, 5, 5, 6, 7, 8, 8, 9, 9, 10, 10, 11, 12, 12, 12, 12, 13, 13, 14, 15, 15, 15, 15, 15, 16, 16, 16, 17, 17, 18, 18, 19, 20, 20, 20, 20, 20, 21, 21, 21, 21, 22, 23, 24, 24, 24, 24, 24, 24, 24, 25, 25, 25, 25, 26, 26, 27, 27, 27, 28, 28, 28, 28, 29, 29, 30, 30, 30, 30, 30, 31, 32, 32, 32

Offset: 1

David W. Wilson

nonn

Sorted union of A009000, A009004, and A009012, retaining duplicates. - Sean A. Irvine, Apr 19 2018

Ron Knott, Pythagorean Triples and Online Calculators

Cf. A009000, A009004, A009012, A009041.

A009185 Long leg of more than one Pythagorean triangle.

Original entry on oeis.org

12, 24, 36, 40, 45, 48, 56, 60, 63, 72, 80, 84, 90, 96, 105, 108, 112, 120, 126, 132, 135, 140, 144, 156, 160, 165, 168, 176, 180, 189, 192, 195, 200, 204, 208, 210, 216, 220, 224, 225, 228, 231, 240, 252, 255, 260, 264, 270, 272, 273, 275, 276, 280, 285, 288, 300, 304

Offset: 1

David W. Wilson

nonn

If n is in the sequence, k*n is in the sequence for all k > 1. So sequence is union of arithmetic progressions such as numbers of the form 12*k, 40*k, 45*k, ... - Altug Alkan, Nov 30 2015

Numbers appearing more than once in A009012. - Sean A. Irvine, Apr 20 2018

Cf. A009012, A009188, A020883, A033942.

cp's OEIS Frontend

A009000 Ordered hypotenuse numbers (squares are sums of 2 distinct nonzero squares).

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A009023 Long legs of Pythagorean triangles.

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A057096 Saint-Exupéry numbers: ordered products of the three sides of Pythagorean triangles.

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A156681 Consider all Pythagorean triangles A^2 + B^2 = C^2 with A < B < C; sequence gives values of B, sorted to correspond to increasing A (A009004).

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A057099 Middle side of a Pythagorean triangle (ordered by the product of the sides).

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A074235 Numbers that cannot be a long leg of an integer right triangle.

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A376900 a(n) is the number of distinct integer-sided right triangles that can be drawn into a square with side length n.

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Maple

A009041 Ordered legs of Pythagorean triangles.

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