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A360946 Number of Pythagorean quadruples with inradius n.

%I A360946 #30 May 11 2023 12:16:32
%S A360946 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 A360946 64,95,116,132,83,153,90,144,149,133,108,238,108,162,169,171,122,284,
%U A360946 130,219,200,196,145,340,174,201,231,239,164,364,176,314,278,256,190,399,195,281,360,330
%N A360946 Number of Pythagorean quadruples with inradius n.
%C A360946 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 A360946 For every positive integer n, there is at least one Pythagorean quadruple with inradius n.
%D A360946 J. M. Blanco Casado, J. M. Sánchez Muñoz, and M. A. Pérez García-Ortega, El Libro de las Ternas Pitagóricas, Preprint 2023.
%H A360946 Miguel-Ángel Pérez García-Ortega, <a href="/A360946/a360946_1.pdf">Pythagorean Quadruples</a> (in Spanish).
%H A360946 Wikipedia, <a href="https://en.wikipedia.org/wiki/Pythagorean_quadruple">Pythagorean quadruple</a>.
%e A360946 For n=1 the a(1)=1 solution is (1,2,2,3).
%e A360946 For n=2 the a(2)=3 solutions are (1,4,8,9), (2,3,6,7) and (2,4,4,6).
%e A360946 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 A360946 n=50;
%t A360946 div={};suc={};A={};
%t A360946 Do[A=Join[A,{Range[1,(1+1/Sqrt[3])q]}],{q,1,n}];
%t A360946 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 A360946 Cf. A096907, A096908, A096909, A096910, A097263, A097264, A097265, A097266, A097267
%Y A360946 Cf. A169580, A225206, A225207, A230543, A330893, A330894, A331365, A335654, A359741
%K A360946 nonn
%O A360946 1,2
%A A360946 _Miguel-Ángel Pérez García-Ortega_, Feb 26 2023