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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 (list; graph; refs; listen; history; text; internal format)
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]]
CROSSREFS
Sequence in context: A198467 A198456 A337602 * A032570 A130483 A115012
KEYWORD
nonn
AUTHOR
STATUS
approved

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Last modified April 24 04:14 EDT 2024. Contains 371918 sequences. (Running on oeis4.)