A120681 -id:A120681 - 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

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

A118905 Sum of legs of Pythagorean triangles (without multiple entries).

Original entry on oeis.org

7, 14, 17, 21, 23, 28, 31, 34, 35, 41, 42, 46, 47, 49, 51, 56, 62, 63, 68, 69, 70, 71, 73, 77, 79, 82, 84, 85, 89, 91, 92, 93, 94, 97, 98, 102, 103, 105, 112, 113, 115, 119, 123, 124, 126, 127, 133, 136, 137, 138, 140, 141, 142, 146, 147, 151, 153, 154, 155, 158, 161, 164, 167, 168, 170, 175, 178, 182, 184, 186, 187, 188

Offset: 1

Giovanni Resta, May 05 2006

nonn

The prime numbers in this sequence define A001132 (see comment in A001132). - Richard Choulet, Dec 16 2008
For the sum of legs of Pythagorean triangles with multiple entries see A198390. - Wolfdieter Lang, May 24 2013
Are these just the positive multiples of A001132? - Charles R Greathouse IV, May 28 2013
For the sum of legs of primitive Pythagorean triangles see A120681. - Wolfdieter Lang, Feb 17 2015
n is in the sequence iff A331671(n) > 0. - Ray Chandler, Feb 26 2020

			7 = 3 + 4 and 3^2 + 4^2 = 5^2.
a(14) = 49 = 7^2 from the primitive Pythagorean triangle (x,y,z) = (9,40,41), and from the non-primitive one 7*(3,4,5); a(42) = 119 = 7*17 from four Pythagorean triangles (39,80,89) and (99,20,181) (both primitive) and 7*(5,12,13), 17*(3,4,5). - _Wolfdieter Lang_, May 24 2013

Seiichi Manyama, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Pythagorean Triple.

Cf. A009096, A118903, A118904, A058529, A001132, A120681, A331671.

[m:m in [2..200]|#[k:k in [1..m-1]|IsSquare(k^2+(m-k)^2)] ne 0]; // Marius A. Burtea, Jul 29 2019

is(n)=my(t=n^2); forstep(i=2-n%2, n-2, 2, if(issquare((t+i^2)/2), return(1))); 0 \\ Charles R Greathouse IV, May 28 2013

More terms from 147 on. - Richard Choulet, Nov 24 2009

Name specified. - Wolfdieter Lang, May 24 2013

A077242 Combined Diophantine Chebyshev sequences A077240 and A077239.

Original entry on oeis.org

5, 7, 23, 37, 133, 215, 775, 1253, 4517, 7303, 26327, 42565, 153445, 248087, 894343, 1445957, 5212613, 8427655, 30381335, 49119973, 177075397, 286292183, 1032071047, 1668633125, 6015350885, 9725506567, 35060034263, 56684406277, 204344854693, 330380931095

Offset: 0

Wolfdieter Lang, Nov 08 2002

a(n)^2 - 8*b(n)^2 = 17, with the companion sequence b(n)= A077241(n).

Because there is only one primitive Pythagorean triangle with sum of the legs L = 17 (see also A120681), namely (5,12,13), all positive solutions (x(n), y(n)) = (a(n), 2*A077241(n)) of the (generalized) Pell equation x^2 - 2*y^2 = +17 satisfy x(n) < 2*y(n), for n >= 1, only 5 = x(0) > 2*y(0) = 4. The proof runs along the same line as the one given in a comment on the L=7 case in A077443. - Wolfdieter Lang, Feb 05 2015

			23 = a(2) = sqrt(8*A077241(2)^2 + 17) = sqrt(8*8^2 + 17)= sqrt(529) = 23.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (0,6,0,-1).

I:=[5,7,23,37]; [n le 4 select I[n] else 6*Self(n-2)-Self(n-4): n in [1..40]]; // Vincenzo Librandi, Feb 18 2014

A077239 = Table[2*ChebyshevT[n+1, 3] + ChebyshevT[n, 3], {n, 0, 12}]; A077240 = Table[ChebyshevT[n+1, 3] + 2*ChebyshevT[n, 3], {n, 0, 12}]; Riffle[A077240, A077239] (* Jean-François Alcover, Dec 19 2013 *)
CoefficientList[Series[(1 - x) (5 + 12 x + 5 x^2)/(1 - 6 x^2 + x^4), {x, 0, 40}], x] (* Vincenzo Librandi, Feb 18 2014 *)

Vec((1-x)*(5+12*x+5*x^2)/(1-6*x^2+x^4) + O(x^50)) \\ Colin Barker, Mar 27 2016

a(2*k) = A077240(k) and a(2*k+1) = A077239(k), k>=0.
G.f.: (1-x)*(5+12*x+5*x^2)/(1-6*x^2+x^4).
a(n) = 6*a(n-2)-a(n-4) for n>3. - Vincenzo Librandi, Feb 18 2014
a(n) = ((6-5*sqrt(2))*(1-sqrt(2))^n - (-1-sqrt(2))^n*(-4+sqrt(2)) + 4*(-1+sqrt(2))^n + sqrt(2)*(-1+sqrt(2))^n + 6*(1+sqrt(2))^n + 5*sqrt(2)*(1+sqrt(2))^n)/4. - Colin Barker, Mar 27 2016

A145906 Concerning hypotenuses of triangles such that the sum of the two legs is a perfect square.

Original entry on oeis.org

9, 19, 27, 33, 57, 51, 51, 73, 89, 83, 107, 99, 139, 129, 137, 123, 129, 187, 187, 163, 177, 171, 209, 257, 201, 233, 267, 227, 251, 337, 243, 321, 313, 307, 297, 289, 291, 387, 411, 363, 347, 393, 339, 379, 369, 363, 417, 401, 393, 491, 499, 473, 593, 449

Offset: 0

Paul Curtz, Oct 23 2008

Last digit is never 5.
Frenicle considers numbers N (apparently the set of A058529 or A120681) and their squares N^2. These have representations N=2*b^2-a^2 = d^2-2*c^2 with d=b+c and N^2 = 2*f^2-e^2 = h^2-2*g^2 with h=f+g. For example N=7 with a=1, b=2, c=1, d=3 and N^2=49 with e=1, f=5, g=4, h=9. The current sequence contains the list of h's.
Apparently the list of N^2 is A089552, the list of a in A143732, the list of b in A147847, the list of e (in different order) in A152910, the list of f (sorted into a different order) in A020882.

			(a,b,c,d,e,f,g,h) = (1,2,1,3,1,5,4,9) with N=7 or  (1,3,2,5,7,13,6,19) with N=17 or (3,4,1,5,7,17,10,27) with N=23 or (1,4,3,7,17,25,8,33) with N=31.

M. de Frenicle, Methode pour trouver la solutions des problemes par les exclusions, in: Divers ouvrages des mathematiques et de physique par messieurs de l'academie royale des sciences, (1693) pp 1-44.

Cf. A144407

A328971 Numerator of the fraction (hypotenuse - difference of legs) / (sum of legs - hypotenuse) of the n-th primitive Pythagorean triangle.

Original entry on oeis.org

2, 3, 5, 4, 7, 7, 5, 9, 6, 7, 9, 11, 11, 7, 8, 9, 11, 13, 8, 13, 15, 13, 9, 10, 11, 12, 15, 10, 17, 11, 12, 15, 13, 17, 19, 11, 17, 13, 19, 17, 19, 12, 13, 14, 21, 15, 19, 16, 21, 13, 14, 23, 19, 16, 23, 17, 21, 14, 25, 23, 16, 17, 25, 21, 23, 15, 19, 16, 17, 18, 23, 27, 25, 19, 20, 16, 17, 23, 18

Offset: 1

S. Brunner, Nov 01 2019

a(n) / A328972(n) should contain all reduced fractions between 1 and sqrt(2) + 1 without duplicates.
a(n) is built from the difference between the length of the hypotenuse (A020882) and the difference between the two legs (A120682) of the n-th primitive Pythagorean triangle.
A328972(n) (denominators) is built from the difference between the sum of the length of the legs (A120681) and the hypotenuse of the n-th primitive Pythagorean triangle.
Then both numbers are divided by their GCD to get the reduced fraction.
All primitive Pythagorean triangles are sorted first on hypotenuse, then on long leg.

			For n=2 we need the 2nd primitive Pythagorean triangle:
5,12,13
^  ^    We calculate the difference between the two small numbers: 12-5=7.
      ^ And to get our numerator we subtract 7 from the hypotenuse length: 13-7=6.
^  ^    Then we calculate the sum of the two small numbers: 5+12=17.
      ^ We subtract 13 from this sum to get the denominator: 17-13=4.
This gives us the fraction 6/4, and in reduced form 3/2.

S. Brunner, Table of n, a(n) for n = 1..10000
S. Brunner, List for n = 0..5000 together with the primitive Pythagorean triangles

Denominators: A328972.

A328972 Denominator of the fraction (hypotenuse - difference of legs) / (sum of legs - hypotenuse) of the n-th primitive Pythagorean triangle.

Original entry on oeis.org

1, 2, 3, 3, 3, 5, 4, 5, 5, 4, 7, 5, 7, 6, 5, 4, 9, 7, 7, 9, 7, 11, 8, 7, 6, 5, 11, 9, 9, 8, 7, 13, 6, 11, 9, 10, 13, 8, 11, 15, 13, 11, 10, 9, 11, 8, 15, 7, 13, 12, 11, 11, 17, 9, 13, 8, 17, 13, 11, 15, 11, 10, 13, 19, 17, 14, 8, 13, 12, 11, 19, 13, 17, 10, 9, 15, 14, 21, 13

Offset: 1

S. Brunner, Nov 01 2019

A328971(n) / a(n) should contain all reduced fractions between 1 and sqrt(2) +  1 without duplicates.
A328971(n) (numerators) is built from the difference between the length of the hypotenuse (A020882) and the difference between the two legs (A120682) of the n-th primitive Pythagorean triangle.
a(n) is built from the difference between the sum of the length of the legs (A120681) and the hypotenuse of the n-th primitive Pythagorean triangle.
Then both numbers are divided by their GCD to get the reduced fraction.
All primitive Pythagorean triangles are sorted first on hypotenuse, then on long leg.

			For n=13 we need the 13th primitive Pythagorean triangle:
36,77,85
^  ^     We calculate the difference between the two small numbers: 77-36=41.
      ^  To get our numerator we subtract 41 from the hypotenuse length: 85-41=44.
^  ^     Then we calculate the sum of the two small numbers: 36+77=113.
      ^  We subtract 85 from this sum to get the denominator: 113-85=28.
This gives us the fraction 44/28 and in reduced form 11/7.

S. Brunner, Table of n, a(n) for n = 1..10000
S. Brunner, List for n = 0..5000 together with the primitive Pythagorean triangles

Numerators: A328971.

A127923 Difference between squares of legs of primitive Pythagorean triangles, sorted (with multiplicity).

Original entry on oeis.org

7, 41, 119, 161, 239, 527, 721, 959, 1081, 1241, 1393, 1519, 2047, 3281, 3479, 3713, 4207, 4633, 4681, 4879, 5593, 6647, 6887, 7327, 8119, 9401, 9641, 10199, 11753, 12121, 12319, 12593, 16999, 19159, 19199, 19873, 20447, 22393, 23359, 24521, 24521

Offset: 1

Lekraj Beedassy, Feb 06 2007

nonn

This is the sorted sequence of all products A120681(i)*A120682(i). - R. J. Mathar, Sep 24 2007
The sequence is conjectural (and may miss entries) because it is generated from a finite list of primitive Pythagorean triangles. The associated lengths in a^2+b^2=c^2 are (a,b)=(3,4), (21,20), (5,12), (15,8), (119,120), (7,24), (55,48), (65,72), (35,12), (45,28), (697,696), (9,40), (33,56), (105,88), (11,60), (63,16), (297,304), (77,36), (91,60), (39,80), (403,396), (133,156), (13,84), (207,224), (4059, 4060), (99,20), (171,140), (85,132), (117,44), (275,252), (15,112), (153,104), (51,140), (555,572), (95,168), (143,24), (17,144), (253, 204), (225,272), (165,52), (1755,1748), (429,460),... with gcd(a, b)=1 and |a^2-b^2| in the sequence. - R. J. Mathar, Sep 24 2007
Confirmed sequence is accurate and complete. Observe that both b-a and b+a must be in A058529. Running through the possible combinations of those values with products below 25000 that produce values of a and b that are legs of primitive Pythagorean triangles confirms list is correct. Note that terms of this sequence must also be in A058529. - Ray Chandler, Apr 11 2010
24521 appears twice in the sequence for (a,b)=(52,165) and (1748,1755). - Ray Chandler, Apr 11 2010

Cf. A058529, A120681, A120682.

More terms from R. J. Mathar, Sep 24 2007

Removed "conjectural" from description by Ray Chandler, Apr 11 2010

cp's OEIS Frontend

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

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

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A118905 Sum of legs of Pythagorean triangles (without multiple entries).

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A077242 Combined Diophantine Chebyshev sequences A077240 and A077239.

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A145906 Concerning hypotenuses of triangles such that the sum of the two legs is a perfect square.

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A328971 Numerator of the fraction (hypotenuse - difference of legs) / (sum of legs - hypotenuse) of the n-th primitive Pythagorean triangle.

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A328972 Denominator of the fraction (hypotenuse - difference of legs) / (sum of legs - hypotenuse) of the n-th primitive Pythagorean triangle.

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A127923 Difference between squares of legs of primitive Pythagorean triangles, sorted (with multiplicity).

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