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A070109 Number of right integer triangles with perimeter n and relatively prime side lengths.

%I A070109 #20 Feb 16 2025 08:32:46
%S A070109 0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,
%T A070109 0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,
%U A070109 0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0
%N A070109 Number of right integer triangles with perimeter n and relatively prime side lengths.
%C A070109 Right integer triangles have integer areas: see A070142, A051516.
%C A070109 a(n) is nonzero iff n is in A024364.
%H A070109 Antti Karttunen, <a href="/A070109/b070109.txt">Table of n, a(n) for n = 1..20000</a> (obtained from the b-file of A078926)
%H A070109 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RightTriangle.html">Right Triangle</a>.
%H A070109 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PythagoreanTriple.html">Pythagorean Triples</a>.
%H A070109 Reinhard Zumkeller, <a href="/A070080/a070080.txt">Integer-sided triangles</a>
%F A070109 a(n) = A078926(n/2) if n is even; a(n)=0 if n is odd.
%F A070109 a(n) = A051493(n) - A070094(n) - A070102(n).
%F A070109 a(n) <= A024155(n).
%e A070109 For n=30 there are A005044(30) = 19 integer triangles; only one is right: 5+12+13 = 30, 5^2+12^2 = 13^2; therefore a(30) = 1.
%t A070109 unitaryDivisors[n_] := Cases[Divisors[n], d_ /; GCD[d, n/d] == 1];
%t A070109 A078926[n_] := Count[unitaryDivisors[n], d_ /; OddQ[d] && Sqrt[n] < d < Sqrt[2n]];
%t A070109 a[n_] := If[EvenQ[n], A078926[n/2], 0];
%t A070109 Table[a[n], {n, 1, 1716}] (* _Jean-François Alcover_, Oct 04 2021 *)
%Y A070109 Cf. A070080, A070081, A070082, A051493, A070093, A070101, A070138, A070084, A070137.
%K A070109 nonn
%O A070109 1,1716
%A A070109 _Reinhard Zumkeller_, May 05 2002
%E A070109 Secondary offset added by _Antti Karttunen_, Oct 07 2017