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A360946
Number of Pythagorean quadruples with inradius n.
1
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
1,2
COMMENTS
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.
REFERENCES
J. M. Blanco Casado, J. M. Sánchez Muñoz, and M. A. Pérez García-Ortega, El Libro de las Ternas Pitagóricas, Preprint 2023.
LINKS
Miguel-Ángel Pérez García-Ortega, Pythagorean Quadruples (in Spanish).
EXAMPLE
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).
MATHEMATICA
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]]
KEYWORD
nonn
STATUS
approved