A046087 -id:A046087 - cp's OEIS frontend

A118962 Difference between short leg and hypotenuse in primitive Pythagorean triangles, sorted on hypotenuse (A020882), then on long leg (A046087).

2, 8, 9, 18, 9, 25, 32, 25, 50, 32, 49, 25, 49, 72, 50, 32, 81, 49, 98, 81, 49, 121, 128, 98, 72, 50, 121, 162, 81, 128, 98, 169, 72, 121, 81, 200, 169, 128, 121, 225, 169, 242, 200, 162, 121, 128, 225, 98, 169, 288, 242, 121, 289, 162, 169, 128, 289, 338, 121, 225, 242

Offset: 1

Lekraj Beedassy, May 07 2006

nonn

Entries take only values appearing in A096033.

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

Cf. A118961, A020882, A046086, A096033, A120682.

a(n) = A020882(n) - A046086(n) = A118961(n) + A120682(n). - Paul Curtz, Dec 11 2008

Corrected and extended by Joshua Zucker, May 11 2006

A118961 Difference between long leg and hypotenuse in primitive Pythagorean triangles, sorted on hypotenuse (A020882), then on long leg (A046087).

Original entry on oeis.org

1, 1, 2, 1, 8, 2, 1, 8, 1, 9, 2, 18, 8, 1, 9, 25, 2, 18, 1, 8, 32, 2, 1, 9, 25, 49, 8, 1, 32, 9, 25, 2, 49, 18, 50, 1, 8, 25, 32, 2, 18, 1, 9, 25, 50, 49, 8, 81, 32, 1, 9, 72, 2, 49, 50, 81, 8, 1, 98, 32, 25, 49, 72, 2, 18, 1, 121, 9, 25, 49, 8, 98, 32, 81, 121, 1, 9, 2, 25, 18, 128, 49, 50

Offset: 1

Lekraj Beedassy, May 07 2006

nonn

Entries take only values appearing in A096033.

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

Cf. A118962, A020882, A046087, A096033, A120682.

a(n) = A020882(n) - A046087(n) = A118962(n) - A120682(n). - Ray Chandler, Nov 24 2019

More terms from Joshua Zucker, May 11 2006

A120734 Area of primitive Pythagorean triangles sorted on hypotenuse (A020882), then on middle side (or long leg A046087).

Original entry on oeis.org

6, 30, 60, 84, 210, 210, 180, 630, 330, 924, 504, 1320, 1386, 546, 1560, 2340, 990, 2730, 840, 2574, 4620, 1716, 1224, 3570, 5610, 7140, 4290, 1710, 7956, 5016, 7980, 2730, 10374, 7854, 11970, 2310, 6630, 10920, 12540, 4080, 11856, 3036, 8970, 14490

Offset: 1

Lekraj Beedassy, Aug 18 2006, Aug 20 2006

nonn

Not the same as A057229.

Michael Somos, Table of primitive Pythagorean triples and related parameters

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

Original entry on oeis.org

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

A058529 Numbers whose prime factors are all congruent to +1 or -1 modulo 8.

Original entry on oeis.org

1, 7, 17, 23, 31, 41, 47, 49, 71, 73, 79, 89, 97, 103, 113, 119, 127, 137, 151, 161, 167, 191, 193, 199, 217, 223, 233, 239, 241, 257, 263, 271, 281, 287, 289, 311, 313, 329, 337, 343, 353, 359, 367, 383, 391, 401, 409, 431, 433, 439, 449, 457, 463, 479, 487

Offset: 1

William Bagby (bagsbee(AT)aol.com), Dec 24 2000

Numbers of the form x^2 - 2*y^2, where x is odd and x and y are relatively prime. - Franklin T. Adams-Watters, Jun 24 2011
Consider primitive Pythagorean triangles (a^2 + b^2 = c^2, gcd(a, b) = 1, a <= b); sequence gives values b-a, sorted with duplicates removed; terms > 1 in sequence give values of a + b, sorted. (See A046086 and A046087.)
Ordered set of (semiperimeter + radius of largest inscribed circle) of all primitive Pythagorean triangles. Semiperimeter of Pythagorean triangle + radius of largest circle inscribed in triangle = ((a+b+c)/2) + ((a+b-c)/2) = a + b.
The terms of this sequence are all of the form 6*N +- 1, since the prime divisors are, and numbers of this form are closed under multiplication. In fact, all terms are == 1, 7, 17, or 23 (mod 24). - J. T. Harrison (harrison_uk_2000(AT)yahoo.co.uk), Apr 28 2009, edited by Franklin T. Adams-Watters, Jun 24 2011
Is similar to A001132, but includes composites whose factors are in A001132. Can be generated in this manner.
Third side of primitive parallepipeds with square base; that is, integer solution of a^2 + b^2 + c^2 = d^2 with gcd(a,b,c) = 1 and b = c. - Carmine Suriano, May 03 2013
Other than -1, values of difference z-y that solve the Diophantine equation x^2 + y^2 = z^2 + 2. - Carmine Suriano, Jan 05 2015
For k > 1, k is in the sequence iff A330174(k) > 0. - Ray Chandler, Feb 26 2020

B Berggren, Pytagoreiska trianglar. Tidskrift för elementär matematik, fysik och kemi, 17:129-139, 1934.
Olaf Delgado-Friedrichs and Michael O’Keeffe, Edge-transitive lattice nets, Acta Cryst. (2009). A65, 360-363.

Ray Chandler, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
F. Barnes, primitive Pythagorean triangles where a-b is a constant.
Johannes Boot, Draft English translation of B Berggren's (1934) article "Pytagoreiska Trianglar", ResearchGate 2017.
K. S. Brown, Pythagorean graphs.
O. Delgado-Friedrichs and M. O'Keeffe, Edge-transitive lattice nets, Acta Cryst. A, A65 (2009), 360-363.
B. Frénicle, Méthode pour trouver la solution des problèmes par les exclusions, 44 pages (see p. 31). In Divers ouvrages de mathematique .. Par Messieurs de l'Academie Royale des Sciences, in-fol, 6+518+1PP, Paris, 1693. - _Paul Curtz_, Sep 06 2008

Cf. A020882-A020886, A020888, A046086, A046087, A014498, A001132, A001653, A027748, A047522, A330174.

a058529 n = a058529_list !! (n-1)
a058529_list = filter (\x -> all (`elem` (takeWhile (<= x) a001132_list))
                                 $ a027748_row x) [1..]
-- Reinhard Zumkeller, Jan 29 2013

Select[Range[500], Union[Abs[Mod[Transpose[FactorInteger[#]][[1]], 8, -1]]] == {1} &] (* T. D. Noe, Feb 07 2012 *)

is(n)=my(f=factor(n)[,1]%8); for(i=1,#f, if(f[i]!=1 && f[i]!=7, return(0))); 1 \\ Charles R Greathouse IV, Aug 01 2016

a(n) = |A-B|=|j^2-2*k^2|, j=(2*n-1), k,n in N, GCD(j,k)=1, the absolute difference between primitive Pythagorean triple legs (sides adjacent to the right angle). - Roger M Ellingson, Dec 09 2023

More terms from Naohiro Nomoto, Jul 02 2001
Edited by Franklin T. Adams-Watters, Jun 24 2011
Duplicated comment removed and name rewritten by Wolfdieter Lang, Feb 17 2015

A046086 Smallest member 'a' of the primitive Pythagorean triples (a,b,c) ordered by increasing c, then b.

Original entry on oeis.org

3, 5, 8, 7, 20, 12, 9, 28, 11, 33, 16, 48, 36, 13, 39, 65, 20, 60, 15, 44, 88, 24, 17, 51, 85, 119, 52, 19, 104, 57, 95, 28, 133, 84, 140, 21, 60, 105, 120, 32, 96, 23, 69, 115, 160, 161, 68, 207, 136, 25, 75, 204, 36, 175, 180, 225, 76, 27, 252, 152, 135, 189

Offset: 1

Eric W. Weisstein

nonn

Ivan Neretin, Table of n, a(n) for n = 1..10000
F. Richman, Pythagorean Triples
Eric Weisstein's World of Mathematics, Pythagorean Triple.

Cf. A046087, A020882.

maxHypo = 389; r[b_, c_] := Reduce[0 < a <= b < c && a^2 + b^2 == c^2, a, Integers]; Reap[Do[r0 = r[b, c]; If[r0 =!= False, {a0, b0, c0} = {a, b, c} /. ToRules[r0]; If[GCD[a0, b0, c0] == 1, Print[a0]; Sow[a0]]], {c, 1, maxHypo}, {b, 1, maxHypo}]][[2, 1]] (* Jean-François Alcover, Oct 22 2012 *)

A096910 Primitive Pythagorean Quadruples a^2+b^2+c^2=d^2, 0

Original entry on oeis.org

3, 7, 9, 9, 11, 11, 13, 15, 15, 17, 17, 19, 19, 19, 21, 21, 21, 21, 23, 23, 23, 25, 25, 27, 27, 27, 27, 27, 29, 29, 29, 31, 31, 31, 31, 33, 33, 33, 33, 33, 33, 33, 35, 35, 35, 35, 37, 37, 37, 37, 39, 39, 39, 39, 39, 39, 41, 41, 41, 41, 41, 43, 43, 43, 43, 43, 43, 45, 45, 45

Offset: 1

Ray Chandler, Aug 15 2004

nonn

Sequence with repetitions removed is A005818. - Ivan Neretin, May 24 2015

Ivan Neretin, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Pythagorean Quadruple.

Cf. A096907, A096908, A096909 (other components of the quadruple), A046086, A046087, A020882 (Pythagorean triples ordered in a similar way).

mx = 50; res = {}; Do[If[GCD[b, c, d] > 1, Continue[]]; If[IntegerQ[a = Sqrt[d^2 - b^2 - c^2]] && a > 0 && a <= b, AppendTo[res, {a, b, c, d}]], {d, mx}, {c, d}, {b, c}]; res[[All, 4]] (* Ivan Neretin, May 24 2015 *)

A120682 Difference between legs of primitive Pythagorean triangles sorted first on hypotenuse, then long leg.

Original entry on oeis.org

1, 7, 7, 17, 1, 23, 31, 17, 49, 23, 47, 7, 41, 71, 41, 7, 79, 31, 97, 73, 17, 119, 127, 89, 47, 1, 113, 161, 49, 119, 73, 167, 23, 103, 31, 199, 161, 103, 89, 223, 151, 241, 191, 137, 71, 79, 217, 17, 137, 287, 233, 49, 287, 113, 119, 47, 281, 337, 23, 193, 217, 151, 97

Offset: 1

Lekraj Beedassy, Jun 24 2006

nonn

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

Cf. A120681, A058529, A046086, A046087, A118961, A118962.

A[s_Integer] := With[{s6 = StringPadLeft[ToString[s], 6, "0"]}, Cases[ Import["https://oeis.org/A" <> s6 <> "/b" <> s6 <> ".txt", "Table"], {, }][[All, 2]]];
A046086 = A@046086;
A046087 = A@046087;
a[n_] := A046087[[n]] - A046086[[n]];
a /@ Range[10000]; (* Jean-François Alcover, Mar 07 2020 *)

a(n) = A046087(n) - A046086(n) = A118962(n) - A118961(n).

Edited and extended by Ray Chandler, Apr 10 2010

A014498 Varying radii of inscribed circles within primitive Pythagorean triples as a function of increasing values of hypotenuse.

Original entry on oeis.org

1, 2, 3, 3, 6, 5, 4, 10, 5, 12, 7, 15, 14, 6, 15, 20, 9, 21, 7, 18, 28, 11, 8, 21, 30, 35, 22, 9, 36, 24, 35, 13, 42, 33, 45, 10, 26, 40, 44, 15, 39, 11, 30, 45, 55, 56, 30, 63, 52, 12, 33, 66, 17, 63, 65, 72, 34, 13, 77, 60, 55, 70, 78, 19, 51, 14, 88, 39, 60, 77, 38, 91, 68, 90

Offset: 1

RALPH PETERSON (ralphp(AT)LIBRARY.NRL.NAVY.MIL)

nonn

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

For ordered values of (a+b-c)/2 see A020888.

Cf. A046086, A046087, A020882, A087459.

Arrange all primitive Pythagorean triples a, b, c by value of hypotenuse c, then by long leg b; for n-th value of c, sequence gives radius of largest inscribed circle, (a+b-c)/2.
a(n) = (A046086(n) + A046087(n) - A020882(n))/2 = A087459(n)/2.
a(n) = sqrt(A118961(n)*A118962(n)/2). - Lekraj Beedassy, May 07 2006

More terms from Asher Auel May 05 2000

Extended by Ray Chandler, Mar 09 2004

A120681 Sum of legs of primitive Pythagorean triangles sorted first on hypotenuse, then long leg.

Original entry on oeis.org

7, 17, 23, 31, 41, 47, 49, 73, 71, 89, 79, 103, 113, 97, 119, 137, 119, 151, 127, 161, 193, 167, 161, 191, 217, 239, 217, 199, 257, 233, 263, 223, 289, 271, 311, 241, 281, 313, 329, 287, 343, 287, 329, 367, 391, 401, 353, 431, 409, 337, 383, 457, 359, 463, 479

Offset: 1

Lekraj Beedassy, Jun 24 2006

nonn

The prime numbers congruent to +1 or -1 modulo 8 of this sequence appear exactly once. For a proof see the W. Lang link under A001132. - Wolfdieter Lang, Feb 17 2015

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

Cf. A120682, A058529, A118905, A198390, A001132, A046086, A046087.

A[s_Integer] := With[{s6 = StringPadLeft[ToString[s], 6, "0"]}, Cases[ Import["https://oeis.org/A" <> s6 <> "/b" <> s6 <> ".txt", "Table"], {, }][[All, 2]]];
A046086 = A@046086;
A046087 = A@046087;
a[n_] := A046087[[n]] + A046086[[n]];
a /@ Range[10000] (* Jean-François Alcover, Mar 07 2020 *)

a(n) = A046087(n) + A046086(n).

Edited and corrected by Ray Chandler, Apr 10 2010

cp's OEIS Frontend

A118962 Difference between short leg and hypotenuse in primitive Pythagorean triangles, sorted on hypotenuse (A020882), then on long leg (A046087).

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A118961 Difference between long leg and hypotenuse in primitive Pythagorean triangles, sorted on hypotenuse (A020882), then on long leg (A046087).

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A120734 Area of primitive Pythagorean triangles sorted on hypotenuse (A020882), then on middle side (or long leg A046087).

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

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A058529 Numbers whose prime factors are all congruent to +1 or -1 modulo 8.

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A046086 Smallest member 'a' of the primitive Pythagorean triples (a,b,c) ordered by increasing c, then b.

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Mathematica

A096910 Primitive Pythagorean Quadruples a^2+b^2+c^2=d^2, 0

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Mathematica

A120682 Difference between legs of primitive Pythagorean triangles sorted first on hypotenuse, then long leg.

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A014498 Varying radii of inscribed circles within primitive Pythagorean triples as a function of increasing values of hypotenuse.