A225207 -id:A225207 - cp's OEIS frontend

A225206 Number of Pythagorean quadruples (a, b, c, d) with d < 10^n.

6, 571, 56268, 5614390, 561232920, 56120665334, 5612026652893, 561202243017532, 56120219419339591

Offset: 1

Arkadiusz Wesolowski, May 01 2013

a(n) ~ Pi*A225207(n)/(1+G), where G is Catalan's constant (A006752).

			a(1) = 6 because there are six solutions (a, b, c, d) as follows: (1, 2, 2, 3), (2, 4, 4, 6), (2, 3, 6, 7), (1, 4, 8, 9), (3, 6, 6, 9), (4, 4, 7, 9) with d < 10.

Wikipedia, Pythagorean quadruple

Cf. A169580, A181786, A225207.

a(n) = Sum_{k=1..10^n-1} A181786(k). - Max Alekseyev, Feb 28 2023

a(4) from Giovanni Resta, May 01 2013

a(5)-a(9) from Max Alekseyev, Feb 28 2023

A360946 Number of Pythagorean quadruples with inradius n.

Original entry on oeis.org

1, 3, 6, 10, 9, 19, 16, 25, 29, 27, 27, 56, 31, 51, 49, 61, 42, 91, 52, 71, 89, 86, 63, 142, 64, 95, 116, 132, 83, 153, 90, 144, 149, 133, 108, 238, 108, 162, 169, 171, 122, 284, 130, 219, 200, 196, 145, 340, 174, 201, 231, 239, 164, 364, 176, 314, 278, 256, 190, 399, 195, 281, 360, 330

Offset: 1

Miguel-Ángel Pérez García-Ortega, Feb 26 2023

nonn

A Pythagorean quadruple is a quadruple (a,b,c,d) of positive integers such that a^2 + b^2 + c^2 = d^2 with a <= b <= c. Its inradius is (a+b+c-d)/2, which is a positive integer.

For every positive integer n, there is at least one Pythagorean quadruple with inradius n.

			For n=1 the a(1)=1 solution is (1,2,2,3).
For n=2 the a(2)=3 solutions are (1,4,8,9), (2,3,6,7) and (2,4,4,6).
For n=3 the a(3)=6 solutions are (1,6,18,19), (2,5,14,15), (2,6,9,11), (3,4,12,13), (3,6,6,9) and (4,4,7,9).

J. M. Blanco Casado, J. M. Sánchez Muñoz, and M. A. Pérez García-Ortega, El Libro de las Ternas Pitagóricas, Preprint 2023.

Miguel-Ángel Pérez García-Ortega, Pythagorean Quadruples (in Spanish).
Wikipedia, Pythagorean quadruple.

Cf. A096907, A096908, A096909, A096910, A097263, A097264, A097265, A097266, A097267

Cf. A169580, A225206, A225207, A230543, A330893, A330894, A331365, A335654, A359741

n=50;
div={};suc={};A={};
Do[A=Join[A,{Range[1,(1+1/Sqrt[3])q]}],{q,1,n}];
Do[suc=Join[suc,{Length[div]}];div={};For [i=1,i<=Length[Extract[A,q]],i++,div=Join[div,Intersection[Divisors[q^2+(Extract[Extract[A,q],i]-q)^2],Range[2(Extract[Extract[A,q],i]-q),Sqrt[q^2+(Extract[Extract[A,q],i]-q)^2]]]]],{q,1,n}];suc=Rest[Join[suc,{Length[div]}]];matriz={{"q"," ","cuaternas"}};For[j=1,j<=n,j++,matriz=Join[matriz,{{j," ",Extract[suc,j]}}]];MatrixForm[Transpose[matriz]]

A379744 Number of primitive Pythagorean quintuples (a, b, c, d, e) with 0 < a <= b <= c <= d <= e <= 10^n.

Original entry on oeis.org

10, 5568, 5302303, 5279762116, 5277410421368, 5277177914347752, 5277147974562930196, 5277145259376056385184, 5277145005746992952994327

Offset: 1

Asif Ahmed, Dec 31 2024

A Pythagorean quintuple (x,y,z,w,u) is a solution to x^2+y^2+z^2+w^2=u^2.

			a(1) = 10 because there are ten primitive solutions (a, b, c, d, e) as follows: (1, 1, 1, 1, 2), (1, 1, 3, 5, 6), (1, 1, 7, 7, 10), (1, 2, 2, 4, 5), (1, 3, 3, 9, 10), (1, 4, 4, 4, 7), (1, 5, 5, 7, 10), (2, 2, 3, 8, 9), (2, 2, 4, 5, 7), and (2, 4, 5, 6, 9) with e <= 10.

Math Stack Exchange, Pythagorean quintuple x^2+y^2+z^2+w^2=u^2 has infinite solutions. Is it difficult to find the general formula of its integer solution?.

Cf. A101931, A157085, A225207.

Limit_{n -> oo} a(n)/ 10^(3*n) = 5/(96*Pi^2) ~ 0.005277144981371758929368722042173314526269...

a(n) ~ 5*10^(3*n)/(96*Pi^2) + (3/A - 1/G)*10^(2*n)/64 + (1/(2*sqrt(3)) - 1/(4*sqrt(2)))*10^n/Pi, where A is the Dirichlet L-function value evaluated at s = 2 for the Dirichlet character with modulus 8 and index 4, and G is the Catalan's constant. (A ~ 1.064734171043503370392827451461668889483, G ~ 0.9159655941772190150546035149323841107741)

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A225206 Number of Pythagorean quadruples (a, b, c, d) with d < 10^n.

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A360946 Number of Pythagorean quadruples with inradius n.

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A379744 Number of primitive Pythagorean quintuples (a, b, c, d, e) with 0 < a <= b <= c <= d <= e <= 10^n.

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