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A382932 a(n) is the altitude of the Pythagorean triangle (A046083(A382931(n)), A046084(A382931(n)), A009000(A382931(n))).

%I A382932 #15 May 09 2025 00:51:15
%S A382932 12,24,36,48,60,72,60,84,96,108,120,132,120,144,156,120,168,180,192,
%T A382932 204,216,228,240,180,252,264,276,240,288,300,168,312,324,240,336,348,
%U A382932 360,372,384,396,420,300,408,360,420,432,444,456,468,480,360,492,504,516
%N A382932 a(n) is the altitude of the Pythagorean triangle (A046083(A382931(n)), A046084(A382931(n)), A009000(A382931(n))).
%C A382932 All terms are divisible by 12. Proof: (Start)
%C A382932 Let (a, b, c) be a primitive Pythagorean triple. Since gcd(a, b, c) = 1, all and only the Pythagorean triples (k*c*a, k*c*b, k*c^2) have an integer altitude h = (k*c*a*k*c*b)/(k*c^2) = k*a*b, where k is a positive integer.
%C A382932 With a = p^2 - q^2 and b = 2*p*q follows h = 2*k*p*q*(p^2 - q^2) = k*2*p*q*(p + q)*(p - q), where p > q > 0, gcd(p,q) = 1 and p or q is even.
%C A382932 It is to show that p*q*(p + q)*(p - q) is divisible by 6. Since p or q is divisible by 2, it remains to show that p*q*(p + q)*(p - q) is divisible by 3.
%C A382932 If 3 is a divisor of p or q, p*q is divisible by 3. If p mod 3 = 1 and q mod 3 = 2 or p mod 3 = 2 and q mod 3 = 1, then p + q is divisible by 3. If p mod 3 = q mod 3 = 1 or p mod 3 = q mod 3 = 2, then p - q is divisible by 3.
%C A382932 It follows that all terms are divisible by 12. (End)
%H A382932 Felix Huber, <a href="/A382932/b382932.txt">Table of n, a(n) for n = 1..10000</a>
%H A382932 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PythagoreanTriple.html">Pythagorean Triple</a>
%F A382932 a(n) = A046083(A382931(n))*A046084(A382931(n))/A009000(A382931(n)).
%e A382932 a(1) = 12 because the Pythagorean triangle (A046083(A382931(1)), A046084(A382931(1)), A009000(A382931(1))) = (A046083(7), A046084(7), A009000(7)) = (15, 20, 25) has the integer altitude 15*20/25 = 12.
%p A382932 A382932:=proc(H) # All hypotenuses <= H.
%p A382932     local a,b,c,k,p,q,L,M;
%p A382932     L:=[];
%p A382932     M:=[];
%p A382932     for p from 2 to floor(sqrt(H-1)) do
%p A382932         for q to min(p-1,floor(sqrt(H-p^2))) do
%p A382932             if gcd(p,q)=1 and is(p-q,odd) then
%p A382932                 a:=p^2-q^2;
%p A382932                 b:=2*p*q;
%p A382932                 c:=p^2+q^2;
%p A382932                 for k to iquo(H,c) do
%p A382932                     L:=[op(L),[k*c,k*max(a,b),k*a*b/c]]
%p A382932                 od
%p A382932             fi
%p A382932         od
%p A382932     od;
%p A382932     L:=sort(L);
%p A382932     for k to nops(L) do
%p A382932         if is(L[k,3],integer) then
%p A382932            M:=[op(M),L[k,3]]
%p A382932         fi
%p A382932     od;
%p A382932     return op(M)
%p A382932 end proc;
%p A382932 A382932(1075);
%Y A382932 Cf. A008594, A009000, A046083, A046084, A382931.
%K A382932 nonn
%O A382932 1,1
%A A382932 _Felix Huber_, Apr 13 2025