A024409 -id:A024409 - cp's OEIS frontend

A020882 Ordered hypotenuses (with multiplicity) of primitive Pythagorean triangles.

5, 13, 17, 25, 29, 37, 41, 53, 61, 65, 65, 73, 85, 85, 89, 97, 101, 109, 113, 125, 137, 145, 145, 149, 157, 169, 173, 181, 185, 185, 193, 197, 205, 205, 221, 221, 229, 233, 241, 257, 265, 265, 269, 277, 281, 289, 293, 305, 305, 313, 317, 325, 325, 337, 349, 353, 365, 365

Offset: 1

Clark Kimberling

The largest member 'c' of the primitive Pythagorean triples (a,b,c) ordered by increasing c.
These are numbers of the form a^2 + b^2 where gcd(b-a, 2*a*b)=1. - M. F. Hasler, Apr 04 2010
Equivalently, numbers of the form a^2 + b^2 where gcd(a,b) = 1 and a and b are not both odd. To avoid double-counting, require a > b > 0. - Franklin T. Adams-Watters, Mar 15 2015
The density of such points in a circle with radius squared = a(n) is ~ Pi * a(n). Restricting to a > b > 0 reduces this by a factor of 1/8; requiring gcd(a,b)=1 provides a factor of 6/Pi^2; and a, b not both odd is a factor of 2/3. (2/3, not 3/4, because the case a, b both even has already been eliminated.) Multiplying, a(n) * Pi * 1/8 * 6/Pi^2 * 2/3 is a(n) / (2 * Pi). But n is approximately this number of points, so a(n) ~ 2 * Pi * n. Conjectured by David W. Wilson, proof by Franklin T. Adams-Watters, Mar 15 2015
Permutations are in A094194, A088511, A121727, A119321, A113482 and A081804. Entries of A024409 occur here more than once. - R. J. Mathar, Apr 12 2010
The distinct terms of this sequence seem to constitute a subset of the sequence defined as a(n) = (-1)^n + 6*n for n >= 1. - Alexander R. Povolotsky, Mar 15 2015
The terms in this sequence are given by f(m,n) = m^2 + n^2 where m and n are any two integers satisfying m > 1, n < m, the greatest common divisor of m and n is 1, and m and n are both not odd. E.g., f(m,n) = f(2,1) = 2^2 + 1^2 = 4 + 1 = 5. - Agola Kisira Odero, Apr 29 2016

M. de Frénicle, "Méthode pour trouver la solutions des problèmes par les exclusions", in: "Divers ouvrages de mathématiques et de physique, par Messieurs de l'Académie royale des sciences", Paris, 1693, pp 1-44.

David W. Wilson, Table of n, a(n) for n = 1..10000 (first 1593 terms from M. F. Hasler)
M. de Frénicle, Méthode pour trouver la solutions des problèmes par les exclusions (B.N.F. permanent link to a scan of the original edition).
Werner Hürlimann, Exact and Asymptotic Evaluation of the Number of Distinct Primitive Cuboids, Journal of Integer Sequences, Vol. 18 (2015), Article 15.2.5.
Hans Isdahl, Pythagoras site (in Norwegian). [from Internet Archive Wayback Machine]
Ron Knott, Pythagorean Triples and Online Calculators.
H. P. Robinson and N. J. A. Sloane, Correspondence, 1971-1972.
E. S. Rowland, Primitive Solutions to x^2 + y^2 = z^2.
Michael Somos, Table of primitive Pythagorean triples and related parameters.
Eric Weisstein's World of Mathematics, Pythagorean Triple.

Cf. A004613, A008846, A020883-A020886, A046086, A046087, A222946 (as a number triangle).

t={};Do[Do[a=Sqrt[c^2-b^2];If[a>b,Break[]];If[IntegerQ[a]&&GCD[a,b,c]==1,AppendTo[t,c]],{b,c-1,3,-1}],{c,400}];t (* Vladimir Joseph Stephan Orlovsky, Jan 21 2012 *)
f[c_] := Block[{a = 1, b, lst = {}}, While[b = Sqrt[c^2 - a^2]; a < b, If[ IntegerQ@ b && GCD[a, b, c] == 1, AppendTo[lst, a]]; a++]; lst]
Join @@ Table[ConstantArray[n, Length@f@n], {n, 1, 400, 4}] (* Robert G. Wilson v, Mar 16 2014; corrected by Andrey Zabolotskiy, Oct 31 2019 *)

{my( c=0, new=[]); for( b=1,99, for( a=1, b-1, gcd(b-a,2*a*b) == 1 && new=concat(new,a^2+b^2)); new=vecsort(new); for( j=1,#new, new[j] > (b+1)^2 & (new=vecextract(new, Str(j,".."))) & next(2); write("b020882.txt",c++," "new[j])); new=[])} \\ M. F. Hasler, Apr 04 2010

a(n) = sqrt((A120681(n)^2 + A120682(n)^2)/2). - Lekraj Beedassy, Jun 24 2006
a(n) = sqrt(A046086(n)^2 + A046087(n)^2). - Zak Seidov, Apr 12 2011
a(n) ~ 2*Pi*n. - observation by David W. Wilson, proved by Franklin T. Adams-Watters (cf. comments), Mar 15 2015
a(n) = sqrt(A180620(n)^2 + A231100(n)^2). - Rui Lin, Oct 09 2019

Edited by N. J. A. Sloane, May 15 2010

A008846 Hypotenuses of primitive Pythagorean triangles.

Original entry on oeis.org

5, 13, 17, 25, 29, 37, 41, 53, 61, 65, 73, 85, 89, 97, 101, 109, 113, 125, 137, 145, 149, 157, 169, 173, 181, 185, 193, 197, 205, 221, 229, 233, 241, 257, 265, 269, 277, 281, 289, 293, 305, 313, 317, 325, 337, 349, 353, 365, 373, 377, 389, 397, 401, 409, 421, 425, 433

Offset: 1

N. J. A. Sloane, Ralph Peterson (RALPHP(AT)LIBRARY.nrl.navy.mil)

Numbers of the form x^2 + y^2 where x is even, y is odd and gcd(x, y)=1. Essentially the same as A004613.
Numbers n for which there is no solution to 4/n = 2/x + 1/y for integers y > x > 0. Related to A073101. - T. D. Noe, Sep 30 2002
Discovered by Frénicle (on Pythagorean triangles): Méthode pour trouver ..., page 14 on 44. First text of Divers ouvrages ... Par Messieurs de l'Académie Royale des Sciences, in-folio, 6+518+1 pp., Paris, 1693. Also A020882 with only one of doubled terms (first: 65). - Paul Curtz, Sep 03 2008
All divisors of terms are of the form 4*k+1 (products of members of A002144). - Zak Seidov, Apr 13 2011
A024362(a(n)) > 0. - Reinhard Zumkeller, Dec 02 2012
Closed under multiplication. Primitive elements are in A002144. - Jean-Christophe Hervé, Nov 10 2013
Not only the square of these numbers is equal to the sum of two nonzero squares, but the numbers themselves also are; this sequence is then a subsequence of A004431. - Jean-Christophe Hervé, Nov 10 2013
Conjecture: numbers p for which sqrt(-1) exists in the p-adic numbering system. For example the 5-adic number ...2431212, when squared, gives ...4444444, which is -1, and 5 is in the sequence. - Thierry Banel, Aug 19 2022
The above conjecture was proven true by George Bergman. 3 known facts: (1) prime factors of a(n) are equal to 1 mod 4, (2) modulo such primes, sqrt(-1) exists, (3) if sqrt(m) exists mod r, r being odd, this extends to sqrt(m) in the r-adic ring. - Thierry Banel, Jul 04 2025

A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, pp. 10, 107.

Zak Seidov, Table of n, a(n) for n = 1..87881 (with a(n) up to 10^6).
Ron Knott, Pythagorean Triples and Online Calculators

Subsequence of A004431 and of A000404 and of A339952; primitive elements: A002144.
Cf. A004613, A020882, A073101.
Cf. A137409 (complement), disjoint union of A024409 and A120960.

a008846 n = a008846_list !! (n-1)
a008846_list = filter f [1..] where
   f n = all ((== 1) . (`mod` 4)) $ filter ((== 0) . (n `mod`)) [1..n]
-- Reinhard Zumkeller, Apr 27 2011

for x from 1 by 2 to 50 do for y from 2 by 2 to 50 do if gcd(x,y) = 1 then print(x^2+y^2); fi; od; od; [ then sort ].

Union[ Map[ Plus@@(#1^2)&, Select[ Flatten[ Array[ {2*#1, 2*#2-1}&, {10, 10} ], 1 ], GCD@@#1 == 1& ] ] ] (* Olivier Gérard, Aug 15 1997 *)
lst = {}; Do[ If[ GCD[m, n] == 1, a = 2 m*n; b = m^2 - n^2; c = m^2 + n^2; AppendTo[lst, c]], {m, 100}, {n, If[ OddQ@m, 2, 1], m - 1, 2}]; Take[ Union@ lst, 57] (* Robert G. Wilson v, May 02 2009 *)
Union[Sqrt[#[[1]]^2+#[[2]]^2]&/@Union[Sort/@({Times@@#,(Last[#]^2-First[#]^2)/2}&/@ (Select[Subsets[Range[1,33,2],{2}],GCD@@#==1&]))]] (* Harvey P. Dale, Aug 26 2012 *)

is(n)=Set(factor(n)[,1]%4)==[1] \\ Charles R Greathouse IV, Nov 06 2015

# for an array from the beginning
from math import gcd, isqrt
hypothenuses_upto = 433
A008846 = set()
for x in range(2, isqrt(hypothenuses_upto)+1):
    for y in range(min(x-1, (yy:=isqrt(hypothenuses_upto-x**2))-(yy%2 == x%2)) , 0, -2):
        if gcd(x,y) == 1: A008846.add(x**2 + y**2)
print(A008846:=sorted(A008846)) # Karl-Heinz Hofmann, Sep 30 2024

# for single k
from sympy import factorint
def A008846_isok(k): return not any([(pf-1) % 4 for pf in factorint(k)]) # Karl-Heinz Hofmann, Oct 01 2024

x^2 + y^2 where x is even, y is odd and gcd(x, y)=1. Essentially the same as A004613.

More terms from T. D. Noe, Sep 30 2002

A024362 Number of primitive Pythagorean triangles with hypotenuse n.

Original entry on oeis.org

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

Offset: 1

David W. Wilson

nonn

Consider primitive Pythagorean triangles (A^2 + B^2 = C^2, (A, B) = 1, A <= B); sequence gives number of times C takes value n.
a(A137409(n)) = 0; a(A008846(n)) > 0; a(A120960(n)) = 1; a(A024409(n)) > 1; a(A159781(n)) = 4. - Reinhard Zumkeller, Dec 02 2012
If the formula given below is used one is sure to find all a(n) values for hypotenuses n <= N if the summation indices r and s are cut off at rmax(N) = floor((sqrt(N-4)+1)/2) and smax(N) = floor(sqrt(N-1)/2). a(n) is the number of primitive Pythagorean triples with hypotenuse n modulo catheti exchange. - Wolfdieter Lang, Jan 10 2016

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

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Ron Knott, Pythagorean Triples and Online Calculators
Eric Weisstein's World of Mathematics, Pythagorean Triple

Cf. A020882, A024361, A046079, A046080.
Cf. A000290, A010052.
Cf. A001221, A002144, A106594.

a024362 n = sum [a010052 y | x <- takeWhile (< nn) $ tail a000290_list,
                             let y = nn - x, y <= x, gcd x y == 1]
            where nn = n ^ 2
-- Reinhard Zumkeller, Dec 02 2012

f:= proc(n) local F;
   F:= numtheory:-factorset(n);
   if map(t -> t mod 4, F) <> {1} then return 0 fi;
   2^(nops(F)-1)
end proc:
seq(f(n),n=1..100); # Robert Israel, Jan 11 2016

Table[a0=IntegerExponent[n,2]; If[n==1 || a0>0, cnt=0, m=n/2^a0; p=Transpose[FactorInteger[m]][[1]]; c=Count[p, _?(Mod[#,4]==1 &)]; If[c==Length[p], cnt=2^(c-1), 0]]; cnt, {n,100}]
a[n_] := If[n==1||EvenQ[n]||Length[Select[FactorInteger[n], Mod[#[[1]], 4]==3 &]] >0, 0, 2^(Length[FactorInteger[n]]-1)]; Array[a, 100] (* Frank M Jackson, Jan 28 2018 *)

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),if(gcd(n,k)==1,m++)));  return(m);} \\ Stanislav Sykora, Mar 23 2015

a(n) = [q^n] T(q), n >= 1, where T(q) = Sum_{r>=1,s>=1} rpr(2*r-1, 2*s)*q^c(r,s), with rpr(k,l) = 1 if gcd(k,l) = 1, otherwise 0, and c(r,s) = (2*r-1)^2 + (2s)^2. - Wolfdieter Lang, Jan 10 2016
If all prime factors of n are in A002144 then a(n) = 2^(A001221(n)-1), otherwise a(n) = 0. - Robert Israel, Jan 11 2016
a(4*n+1) = A106594(n), other terms are 0. - Andrey Zabolotskiy, Jan 21 2022

A120960 Pythagorean prime powers.

Original entry on oeis.org

5, 13, 17, 25, 29, 37, 41, 53, 61, 73, 89, 97, 101, 109, 113, 125, 137, 149, 157, 169, 173, 181, 193, 197, 229, 233, 241, 257, 269, 277, 281, 289, 293, 313, 317, 337, 349, 353, 373, 389, 397, 401, 409, 421, 433, 449, 457, 461, 509, 521, 541, 557, 569, 577, 593

Offset: 1

Lekraj Beedassy, Jul 19 2006

nonn

1 + sum of the indices of the first two numbers in A001844 that are divisible by n, if 1 + the sum of those indices equals n. - Mats Granvik, Oct 16 2007
R. J. Turyn proved [Baliga, et al., p. 129, gives the reference] that Williamson Hadamard matrices exist for 4t = 2(p^k + 1), for all primes p such that p == 1 (mod 4). - L. Edson Jeffery, Apr 10 2012
A024362(a(n)) = 1. - Reinhard Zumkeller, Dec 02 2012

			A001844(1) = 5 is divisible by 5, A001844(3) = 25 is divisible by = 5 and 1+3+1=5, so 5 is a member.
A001844(2) = 13 is divisible by = 13, A001844(10) = 221 is divisible by = 13 and 2+10+1=13 so 13 is a member.

Ray Chandler, Table of n, a(n) for n = 1..10000 (first 1000 terms from Reinhard Zumkeller)
A. Baliga and K. J. Horadam, Cocyclic Hadamard matrices over Z_t X Z^2_2, Australas. J. Combin. 11(1995), 123-134.
Eric Weisstein's World of Mathematics, MathWorld, Centered Square Number.

Cf. Disjoint union of A002144 and A146945.
Cf. A001844, subsequence of A000961.
Cf. A024409, subsequence of A008846.

Generate the indices with: =if(mod(1+2*row()*(row()+1);4*column()+1)=0;row();") Then sum the first two indices if it equals the column + 1. - Mats Granvik, Oct 16 2007

import Data.List (elemIndices)
a120960 n = a120960_list !! (n-1)
a120960_list = map (+ 1) $ elemIndices 1 a024362_list
-- Reinhard Zumkeller, Dec 02 2012

A146945 Hypotenuses of primitive Pythagorean triples which are not prime numbers and which are the hypotenuse of a unique triangle.

Original entry on oeis.org

25, 125, 169, 289, 625, 841, 1369, 1681, 2197, 2809, 3125, 3721, 4913, 5329, 7921, 9409, 10201, 11881, 12769, 15625, 18769, 22201, 24389, 24649, 28561, 29929, 32761, 37249, 38809, 50653, 52441, 54289, 58081, 66049, 68921, 72361, 76729, 78125

Offset: 1

John Harrison (harrison_uk_2000(AT)yahoo.co.uk), Apr 20 2009

nonn

Each term is a prime power of the form p^e where p is in A002144 and e>1.
A proper subset of A120960 by eliminating A002144.
A proper subset of A120961 by eliminating A024409.
A proper subset of A008846 by eliminating A002144 and A024409.
A proper subset of A020882 by eliminating A002144, A024409 and duplicate entries.

Ray Chandler, Table of n, a(n) for n = 1..10000

Cf. A020882, A008846, A002144, A024409, A120960, A120961.

lst1 = {1, 1}; lst2 = {}; Do[ If[ GCD[m, n] == 1, a = 2m*n; b = m^2 - n^2; c = m^2 + n^2; If[ !PrimeQ@c, AppendTo[lst1, c]]], {m, 3, 1000}, {n, If[OddQ@m, 2, 1], m - 1, 2}]; lst1 = Sort@ lst1; Do[ If[ lst1[[n - 1]] != lst1[[n]] && lst1[[n]] != lst1[[n + 1]], AppendTo[lst2, lst1[[n]]]], {n, 2, Length@ lst1 - 1}]; Take[lst2, 50] (* Robert G. Wilson v, May 02 2009 *)

a(7) corrected by and a(17) and further terms from Robert G. Wilson v, May 02 2009

Minor edits to comments. - Ray Chandler, Nov 27 2019

A120961 Composite hypotenuses of primitive Pythagorean triangles.

Original entry on oeis.org

25, 65, 85, 125, 145, 169, 185, 205, 221, 265, 289, 305, 325, 365, 377, 425, 445, 481, 485, 493, 505, 533, 545, 565, 625, 629, 685, 689, 697, 725, 745, 785, 793, 841, 845, 865, 901, 905, 925, 949, 965, 985, 1025, 1037, 1073, 1105, 1145, 1157, 1165, 1189, 1205

Offset: 1

Lekraj Beedassy, Jul 19 2006

nonn

Composite entries of A008846. Disjoint union of A024409 and A146945.

Ray Chandler, Table of n, a(n) for n = 1..10000
César Aguilera, On Pythagorean Triples, hal-02151737 preprint [math.NT], 2019.

Cf. A120960, A008846, A146945, A024409.

lst = {}; Do[ If[ GCD[m, n] == 1, a = 2 m*n; b = m^2 - n^2; c = m^2 + n^2; If[ !PrimeQ@c, AppendTo[lst, c]]], {m, 3, 300}, {n, If[ OddQ@m, 2, 1], m - 1, 2}]; Take[ Union@ lst, 51] (* Robert G. Wilson v, May 02 2009 *)

Term 485, which satisfies 485^2 = 476^2 + 93^2, added by Robert G. Wilson v, May 02 2009

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.

A159781 Values of hypotenuse of primitive Pythagorean triples which can have four different shapes (that is, four different sets of "legs").

Original entry on oeis.org

1105, 1885, 2405, 2465, 2665, 3145, 3445, 3485, 3965, 4505, 4745, 5185, 5365, 5525, 5785, 5945, 6205, 6305, 6409, 6565, 7085, 7345, 7565, 7585, 7685, 8177, 8245, 8585, 8845, 8905, 9061, 9265, 9425, 9605, 9685, 9805, 10205, 10585, 10865

Offset: 1

John T. Harrison (harrison_uk_2000(AT)yahoo.co.uk), Apr 22 2009

nonn

This is a subsequence of A024409, which lists hypotenuse values common to more than one primitive Pythagorean triple. A024409(1) = A006278(2) = 65 is the smallest hypotenuse common to exactly two primitive Pythagorean triples; a(1) = A006278(3) = 1105 is the smallest that is common to four. [edited by Jon E. Schoenfield, Aug 19 2018]

A024362(a(n)) = 4. - Reinhard Zumkeller, Dec 02 2012

Ray Chandler, Table of n, a(n) for n = 1..10000 (first 100 terms from Reinhard Zumkeller)

Cf. A024409 and A146945.

Cf. A006278 (8, 16, etc. shapes). - R. J. Mathar, Apr 12 2010

import Data.List (elemIndices)
a159781 n = a159781_list !! (n-1)
a159781_list = map (+ 1) $ elemIndices 4 a024362_list
-- Reinhard Zumkeller, Dec 02 2012

f[c_] := f[c] = Block[{a = 1, b, cnt = 0, lmt = Floor[Sqrt[c^2/2]]}, While[b = Sqrt[c^2 - a^2]; a < lmt, If[IntegerQ@ b && GCD[a, b, c] == 1, cnt++]; a++]; cnt]Select[1 + 4 Range[2800], f@# > 2 &] (* Robert G. Wilson v, Mar 16 2014 *)

6429 replaced by 6409 and 3 terms added by R. J. Mathar, Apr 12 2010

Missing 8585 and 8845 inserted by Reinhard Zumkeller, Dec 02 2012

A111105 Largest member z of a triple 0

Original entry on oeis.org

697, 925, 1073, 1105, 1394, 1850, 2091, 2146, 2165, 2210, 2665, 2775, 2788, 3219, 3277, 3315, 3485, 3700, 3965, 4181, 4182, 4225, 4292, 4330, 4420, 4453, 4625, 4879, 5330, 5365, 5525, 5550, 5576, 6005, 6273, 6438, 6475, 6495, 6554, 6630, 6970, 7085

Offset: 1

R. J. Mathar, Apr 20 2008

nonn

Subset of A024409. If only primitive triples with gcd(x,y,z)=1 are admitted, the sequence reduces to A137559.

			a(1)=697 represents the (z,y,x)-triples (697,185,153) and (697,680,672).
a(4)=1105 represents the triples (1105,520,264), (1105,561,264), (1105,1073,952) and (1105,1073,975).

Robin Visser, Table of n, a(n) for n = 1..10000
R. A. Beuregard and E. R. Suryanarayan, Pythagorean Boxes, Math. Mag. vol 74 no 3 (2001) pp 222-227.
J. Fricke, On Heron simplices and integer embedding, arXiv:math/0112239 [math.NT], 2001.
R. Hartshorne and Ronald van Luijk, Non-Euclidean Pythagorean triples, a problem of Euler and rational points on K3 surfaces, arXiv:math/0606700 [math.NT], 2006.

Cf. A137559.

A179271 Odd long legs `B` of more than one primitive Pythagorean triangle.

Original entry on oeis.org

2145, 3315, 3465, 4095, 4845, 5005, 5865, 6435, 6545, 6555, 7735, 8645, 9009, 9945, 10005, 10695, 11305, 11781, 13167, 13485, 13685, 13923, 14535, 15015, 15295, 15561, 16065, 16095, 17017, 17205, 17255, 17835, 17955, 18837, 19019, 19065

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jul 06 2010

nonn

2145,752,2273;2145,1568,2657;;3315,812,3413;3315,2852,4373;

Ray Chandler, Table of n, a(n) for n = 1..10000

Cf. A024407, A024408, A024409, A024410, A024411, A161962, A174830

lst1={};lst={0};Do[Do[If[GCD[a,b]==1,c=Sqrt[a^2+b^2];If[IntegerQ[c],AppendTo[lst,b];L=Length[lst];If[lst[[L]]==lst[[L-1]]&&OddQ[lst[[L]]],Print[lst[[L]]];AppendTo[lst1,lst[[L]]]]]],{a,b-1,3,-1}],{b,4,4*7!}];lst1

a(20) - a(36) Robert G. Wilson v, Jul 12 2010

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A020882 Ordered hypotenuses (with multiplicity) of primitive Pythagorean triangles.

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A008846 Hypotenuses of primitive Pythagorean triangles.

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A024362 Number of primitive Pythagorean triangles with hypotenuse n.

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A120960 Pythagorean prime powers.

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A146945 Hypotenuses of primitive Pythagorean triples which are not prime numbers and which are the hypotenuse of a unique triangle.

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A120961 Composite hypotenuses of primitive Pythagorean triangles.

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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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