A020882 -id:A020882 - cp's OEIS frontend

A156679 Consider primitive Pythagorean triangles (A^2 + B^2 = C^2, gcd (A, B) = 1, A < BA020884(n)).

5, 13, 25, 17, 41, 61, 37, 85, 113, 65, 145, 181, 29, 101, 221, 265, 145, 313, 365, 53, 197, 421, 481, 257, 65, 545, 613, 85, 325, 685, 89, 761, 401, 841, 925, 125, 485, 1013, 1105, 73, 577, 1201, 149, 1301, 173, 677, 1405, 1513, 785, 185, 1625, 1741, 109, 229

Offset: 1

Ant King, Feb 15 2009

The ordered sequence of A values is A020884(n) and the ordered sequence of C values is A020882(n) (allowing repetitions) and A008846(n) (excluding repetitions).

			As the first four primitive Pythagorean triples (ordered by increasing A) are (3,4,5), (5,12,13), (7,24,25) and (8,15,17), then a(1)=5, a(2)=13, a(3)=25 and a(4)=17.

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

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
Ron Knott, Right-angled Triangles and Pythagoras' Theorem

Cf. A020884, A020882, A008846, A156678, A156682.

Cf. A037213.

a156679 n = a156679_list !! (n-1)
a156679_list = f 1 1 where
   f u v | v > uu `div` 2        = f (u + 1) (u + 2)
         | gcd u v > 1 || w == 0 = f u (v + 2)
         | otherwise             = w : f u (v + 2)
         where uu = u ^ 2; w = a037213 (uu + v ^ 2)
-- Reinhard Zumkeller, Nov 09 2012

PrimitivePythagoreanTriplets[n_]:=Module[{t={{3,4,5}},i=4,j=5},While[iHarvey P. Dale, May 10 2020 *)

A176256 Numbers of the form 4k+1 with least prime divisor of the form 4m-1.

Original entry on oeis.org

9, 21, 33, 45, 49, 57, 69, 77, 81, 93, 105, 117, 121, 129, 133, 141, 153, 161, 165, 177, 189, 201, 209, 213, 217, 225, 237, 249, 253, 261, 273, 285, 297, 301, 309, 321, 329, 333, 341, 345, 357, 361, 369, 381, 393, 405, 413, 417, 429, 437, 441, 453, 465, 469

Offset: 1

Vladimir Shevelev, Apr 13 2010

nonn

By definition, all terms are composite numbers.

Cannot be the hypotenuse of a primitive Pythagorean triangle. - Robert G. Wilson v, Mar 16 2014

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A176255, A002148, A002145, A016813, A004767.

Complement of A020882 in 1 == Mod 4.

fQ[n_] := Mod[ n, 4] == 1 && Mod[ FactorInteger[n][[1, 1]], 4] == 3; Select[Range@470, fQ] (* Robert G. Wilson v, Apr 08 2014 *)

isok(n) = ((n % 4) == 1) && (f = factor(n)) && ((f[1, 1] % 4) == 3); \\ Michel Marcus, Mar 16 2014

More terms from Michel Marcus, Mar 16 2014

A235598 Begin with a(0) = 3. Let a(n) for n > 0 be the smallest positive integer not yet in the sequence which forms part of a Pythagorean triple when paired with a(n-1).

Original entry on oeis.org

3, 4, 5, 12, 9, 15, 8, 6, 10, 24, 7, 25, 20, 16, 30, 18, 80, 39, 36, 27, 45, 28, 21, 29, 420, 65, 33, 44, 55, 48, 14, 50, 40, 32, 60, 11, 61, 1860, 341, 541, 146340, 15447, 20596, 25745, 32208, 2540, 1524, 635, 381, 508, 16125, 4515, 936, 75, 72, 54, 90, 56

Offset: 0

Jack Brennen, Dec 26 2013

Is the sequence infinite?  Can it "paint itself into a corner" at any point? Note that picking any starting point >= 5 seems to lead to a finite sequence ending in 5,3,4. For example, starting with 6 we get 6,8,10,24,7,25,15,9,12,5,3,4, stop (A235599).
By beginning with 3 or 4, we make sure that the 5,3,4 dead-end is never available.
If infinite, is it a permutation of the integers >= 3? This seems likely.  Proving it doesn't seem easy though.
Comment from Jim Nastos, Dec 30 2013: Your question about whether the sequence can 'paint itself into a corner' is essentially asking if the Pythagorean graph has a Hamiltonian path. As far as I know, the questions in the Cooper-Poirel paper (see link) are still unanswered. They ask whether the graph is k-colorable with a finite k, or whether it is even connected (sort of equivalent to your question of whether it is a permutation of the integers >=3).
Lars Blomberg has computed the sequence out to 3 million terms without finding a dead end.
Position of k>2: 0, 1, 2, 7, 10, 6, 4, 8, 35, 3, 67, 30, 5, 13, 89, 15, 143, 12, 22, 118, 385, 9, 11, ..., see A236243. - Robert G. Wilson v, Jan 17 2014

Robert G. Wilson v, Table of n, a(n) for n = 0..10000
J. Cooper and C. Poirel, Note on the Pythagorean Triple System

Cf. A020882, A020883, A020884, A235599, A236243, A236244.

f[s_List] := Block[{n = s[[-1]]}, sol = Solve[ x^2 + y^2 == z^2 && x > 0 && y > 0 && z > 0 && (x == n || z == n), {x, y, z}, Integers]; Append[s, Min[ Complement[ Union[ Extract[sol, Position[ sol, Integer]]], s]]]]; lst = Nest[f, {3}, 250] (* _Robert G. Wilson v, Jan 17 2014 *)

A094194 Hypotenuses x^2 + y^2 of primitive Pythagorean triangles, sorted on values x of the generator pair (x, y), x>y.

Original entry on oeis.org

5, 13, 17, 25, 29, 41, 37, 61, 53, 65, 85, 65, 73, 89, 113, 85, 97, 145, 101, 109, 149, 181, 125, 137, 157, 185, 221, 145, 169, 193, 265, 173, 185, 205, 233, 269, 313, 197, 205, 221, 277, 317, 365, 229, 241, 289, 421, 257, 265, 281, 305, 337, 377, 425, 481, 293

Offset: 1

Lekraj Beedassy, May 25 2004

nonn

For ordered hypotenuses of primitive Pythagorean triangles see A020882.

The hypotenuse Z of the primitive Pythagorean triple (X, Y, Z) with Xy (x and y coprime and not both odd) using the relation Z = x^2 + y^2. The even leg is 2*x*y and the odd leg is x^2 - y^2. [From Lekraj Beedassy, May 06 2010]

Alfred S. Posamentier, Math Charmers, Tantalizing Tidbits for the Mind, Prometheus Books, NY, 2003, page 145.

Ray Chandler, Table of n, a(n) for n = 1..10000
F. Richman, Pythagorean Triples

Cf. A094192, A094193, A120097, A120098.

Inserted a sqrt(.) operation in the definition - R. J. Mathar, Apr 12 2010

Deleted incorrect sqrt in definition (based on author's initial comment) - Aaron Kastel, Oct 30 2012

A155176 Perimeter s/6 (divided by 6) of primitive Pythagorean triangles such that perimeters are Averages of twin prime pairs, q=p+1, a=q^2-p^2, c=q^2+p^2, b=2pq, s=a+b+c, s-+1 are primes.

Original entry on oeis.org

2, 5, 40, 77, 287, 590, 1335, 1717, 2882, 3337, 3927, 4030, 6902, 7315, 7740, 8932, 15965, 20592, 26070, 27405, 34277, 34580, 40920, 50692, 92132, 96647, 113575, 139690, 160557, 167167, 220225, 237407, 279720, 300832, 310765, 336777, 389895

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jan 21 2009

nonn

p=1,q=2,a=3,b=4,c=5,s=12-+1primes, ...

Cf. A020882, A020886, A020884, A020883, A024364, A024406, A155171, A155173, A155174, A155175

lst={};Do[p=n;q=p+1;a=q^2-p^2;c=q^2+p^2;b=2*p*q;s=a+b+c;If[PrimeQ[s-1]&&PrimeQ[s+1],AppendTo[lst,s/6]],{n,8!}];lst

A180620 Odd legs of primitive Pythagorean triples (with multiplicity) sorted with respect to increasing hypotenuse.

Original entry on oeis.org

3, 5, 15, 7, 21, 35, 9, 45, 11, 63, 33, 55, 77, 13, 39, 65, 99, 91, 15, 117, 105, 143, 17, 51, 85, 119, 165, 19, 153, 57, 95, 195, 187, 133, 171, 21, 221, 105, 209, 255, 247, 23, 69, 115, 231, 161, 285, 273, 207, 25, 75, 323, 253, 175, 299, 225, 357, 27, 275, 345, 135, 189, 325

Offset: 1

Jonathan Vos Post, Sep 12 2010

The primary key is the increasing length of the hypotenuse, A020882. If there is more than one solution with that hypotenuse, the (secondary) sorting key is the even leg.

Only the odd legs 'a' of reduced triangles with gcd(a,b,c)=1, a^2+b^2=c^2, a=q^2-p^2, b=2*p*q, c=q^2+p^2, gcd(p,q)=1 are listed.

			a(1) = 3 because the only triangle with the least possible hypotenuse 5 has catheti 3 and 4.

K. G. Stier, Table of n, a(n) for n = 1..1593
Michael Somos, Pythagorean Triple Table, Reduced integer right triangles, Feb 28, 1998.
Wikipedia, Pythagorean Triple.
Index entries related to Pythagorean Triples.

Cf. A046086, A046087, A020882, A081874, A231100.

Comment on sorting added, more terms appended by R. J. Mathar, Oct 15 2010

Sequence's name and comments corrected by K. G. Stier, Nov 03 2013

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

A155177 Area ar/6 (divided by 6) of primitive Pythagorean triangles such that perimeters are Averages of twin prime pairs, q=p+1, a=q^2-p^2, c=q^2+p^2, b=2pq, ar=a*b/2; s=a+b+c, s-+1 are primes.

Original entry on oeis.org

1, 5, 140, 385, 2870, 8555, 29370, 42925, 93665, 116795, 149226, 155155, 348551, 380380, 414090, 513590, 1229305, 1801800, 2567895, 2767905, 3873301, 3924830, 5053620, 6970150, 17090486, 18362930, 23396450, 31919165, 39336465, 41791750

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jan 21 2009

nonn

p=1,q=2,a=3,b=4,c=5, ar=3*4/2=6, s=12-+1primes, ...

Cf. A020882, A020886, A020884, A020883, A024364, A024406, A155171, A155173, A155174, A155175, A155176

lst={};Do[p=n;q=p+1;a=q^2-p^2;c=q^2+p^2;b=2*p*q;s=a+b+c;ar=a*b/2;If[PrimeQ[s-1]&&PrimeQ[s+1],AppendTo[lst,ar/6]],{n,8!}];lst

A156682 Consider all Pythagorean triangles A^2 + B^2 = C^2 with AA009004(n)).

Original entry on oeis.org

5, 13, 10, 25, 17, 15, 41, 26, 61, 20, 37, 85, 50, 25, 39, 113, 34, 65, 145, 30, 82, 181, 29, 52, 101, 35, 75, 221, 122, 265, 40, 51, 74, 145, 65, 313, 170, 45, 123, 365, 53, 100, 197, 421, 50, 78, 226, 481, 68, 130, 257, 55, 65, 183, 545, 290, 91, 125, 613, 60, 85, 111

Offset: 1

Ant King, Feb 17 2009

The corresponding sequence for primitive triples is A156679. For all triples, the ordered sequence of C values is A020882 (allowing repetitions) and A009003 (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)=5, a(2)=13, a(3)=10 and a(4)=25.

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

Ron Knott, Right-angled Triangles and Pythagoras' Theorem

Cf. A009004, A156681, A156679, A020882, A009003.

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

a(n) = sqrt(A009004(n)^2 + A156681(n)^2).

A198440 Square root of second term of a triple of squares in arithmetic progression that is not a multiple of another triple in (A198384, A198385, A198386).

Original entry on oeis.org

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

Offset: 1

Reinhard Zumkeller, Oct 25 2011

nonn

This sequence gives the hypotenuses of primitive Pythagorean triangles (with multiplicities) ordered according to nondecreasing values of the leg sums x+y (called w in the Zumkeller link, given by A198441). See the comment on the equivalence to primitive Pythagorean triangles in A198441. For the values of these hypotenuses ordered nondecreasingly see A020882. See also the triangle version A222946. - Wolfdieter Lang, May 23 2013

			From _Wolfdieter Lang_, May 22 2013: (Start)
Primitive Pythagorean triangle (x,y,z), even y, connection:
a(8) = 61 because the leg sum x+y = A198441(8) = 71 and due to A198439(8) = 49 one has y = (71+49)/2 = 60 is even, hence x = (71-49)/2 = 11 and z = sqrt(11^2 + 60^2) = 61. (End)

Ray Chandler, Table of n, a(n) for n = 1..10000
Keith Conrad, Arithmetic progressions of three squares
Reinhard Zumkeller, Table of initial values

Cf. A020882, A222946.

a198440 n = a198440_list !! (n-1)
a198440_list = map a198389 a198409_list

wmax = 1000;
triples[w_] := Reap[Module[{u, v}, For[u = 1, u < w, u++, If[IntegerQ[v = Sqrt[(u^2 + w^2)/2]], Sow[{u, v, w}]]]]][[2]];
tt = Flatten[DeleteCases[triples /@ Range[wmax], {}], 2];
DeleteCases[tt, t_List /; GCD@@t > 1 && MemberQ[tt, t/GCD@@t]][[All, 2]] (* Jean-François Alcover, Oct 22 2021 *)

A198436(n) = a(n)^2; a(n) = A198389(A198409(n)).

cp's OEIS Frontend

A156679 Consider primitive Pythagorean triangles (A^2 + B^2 = C^2, gcd (A, B) = 1, A < BA020884(n)).

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A176256 Numbers of the form 4k+1 with least prime divisor of the form 4m-1.

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A235598 Begin with a(0) = 3. Let a(n) for n > 0 be the smallest positive integer not yet in the sequence which forms part of a Pythagorean triple when paired with a(n-1).

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A094194 Hypotenuses x^2 + y^2 of primitive Pythagorean triangles, sorted on values x of the generator pair (x, y), x>y.

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Extensions

A155176 Perimeter s/6 (divided by 6) of primitive Pythagorean triangles such that perimeters are Averages of twin prime pairs, q=p+1, a=q^2-p^2, c=q^2+p^2, b=2*p*q, s=a+b+c, s-+1 are primes.

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Mathematica

A180620 Odd legs of primitive Pythagorean triples (with multiplicity) sorted with respect to increasing hypotenuse.

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Crossrefs

Extensions

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

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Mathematica

Extensions

A155177 Area ar/6 (divided by 6) of primitive Pythagorean triangles such that perimeters are Averages of twin prime pairs, q=p+1, a=q^2-p^2, c=q^2+p^2, b=2*p*q, ar=a*b/2; s=a+b+c, s-+1 are primes.

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A156682 Consider all Pythagorean triangles A^2 + B^2 = C^2 with AA009004(n)).

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A155176 Perimeter s/6 (divided by 6) of primitive Pythagorean triangles such that perimeters are Averages of twin prime pairs, q=p+1, a=q^2-p^2, c=q^2+p^2, b=2pq, s=a+b+c, s-+1 are primes.

A155177 Area ar/6 (divided by 6) of primitive Pythagorean triangles such that perimeters are Averages of twin prime pairs, q=p+1, a=q^2-p^2, c=q^2+p^2, b=2pq, ar=a*b/2; s=a+b+c, s-+1 are primes.