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A328712 Number of non-primitive Pythagorean triples with hypotenuse n.

%I A328712 #20 Jun 20 2020 02:39:23
%S A328712 0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1,1,0,0,0,1,0,0,0,1,
%T A328712 1,0,0,0,1,1,0,0,0,0,1,0,0,0,0,2,1,1,0,0,1,0,0,1,0,1,0,0,0,0,2,0,0,1,
%U A328712 0,1,0,0,0,1,2,0,0,1,0,1,0,1,0,0,2,0,1,0,0,1,1,0,0,0,1,0,0,0,0,2,0,1
%N A328712 Number of non-primitive Pythagorean triples with hypotenuse n.
%C A328712 Pythagorean triple including primitive ones and non-primitive ones. For a certain n, it may be the hypotenuse in either primitive Pythagorean triple, or non-primitive Pythagorean triple, or both.
%C A328712 This sequence is the count of n as hypotenuse in non-primitive Pythagorean triple.
%D A328712 A. Beiler, Recreations in the Theory of Numbers. New York: Dover Publications, pp. 116-117, 1966.
%H A328712 Robert Israel, <a href="/A328712/b328712.txt">Table of n, a(n) for n = 1..10000</a>
%F A328712 a(n) = A046080(n) - A024362(n).
%e A328712 n=5 as hypotenuse in only one primitive Pythagorean triple, (3,4,5); so a(5)=0.
%e A328712 n=10 as hypotenuse in only one non-primitive Pythagorean triple, (6,8,10); so a(10)=1.
%e A328712 n=25 as hypotenuse in one primitive Pythagorean triple (7,24,25) and in one non-primitive Pythagorean triple (15,20,25); so a(25)=1.
%p A328712 f:= proc(n) local R;
%p A328712 if isprime(n) then return 0 fi;
%p A328712   R:= map(t -> subs(t,[x,y]),[isolve(x^2+y^2=n^2)]);
%p A328712   nops(select(t -> t[1]>=1 and t[2]>=t[1] and igcd(t[1],t[2])>1, R))
%p A328712 end proc:
%p A328712 map(f, [$1..100]); # _Robert Israel_, Oct 31 2019
%t A328712 a[n_] := Module[{R, x, y}, If[PrimeQ[n], 0, R = Solve[GCD[x, y] > 1 && x >= 1 && y >= x && x^2 + y^2 == n^2, {x, y}, Integers]; Length[R]]];
%t A328712 Array[a, 102] (* _Jean-François Alcover_, Jun 20 2020, after Maple *)
%Y A328712 Cf. A046080, A024362.
%K A328712 nonn
%O A328712 1,50
%A A328712 _Rui Lin_, Oct 26 2019