A360946 - OEIS

%I #30 May 11 2023 12:16:32

%S 1,3,6,10,9,19,16,25,29,27,27,56,31,51,49,61,42,91,52,71,89,86,63,142,

%T 64,95,116,132,83,153,90,144,149,133,108,238,108,162,169,171,122,284,

%U 130,219,200,196,145,340,174,201,231,239,164,364,176,314,278,256,190,399,195,281,360,330

%N Number of Pythagorean quadruples with inradius n.

%C 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.

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

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

%H Miguel-Ángel Pérez García-Ortega, <a href="/A360946/a360946_1.pdf">Pythagorean Quadruples</a> (in Spanish).

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Pythagorean_quadruple">Pythagorean quadruple</a>.

%e For n=1 the a(1)=1 solution is (1,2,2,3).

%e For n=2 the a(2)=3 solutions are (1,4,8,9), (2,3,6,7) and (2,4,4,6).

%e 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).

%t n=50;

%t div={};suc={};A={};

%t Do[A=Join[A,{Range[1,(1+1/Sqrt[3])q]}],{q,1,n}];

%t 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]]

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

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

%K nonn

%O 1,2

%A _Miguel-Ángel Pérez García-Ortega_, Feb 26 2023