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A382931 Numbers k for which the Pythagorean triangle (A046083(k), A046084(k), A009000(k)) has an integer altitude.

%I A382931 #9 Apr 19 2025 17:48:30
%S A382931 7,19,36,51,69,88,99,106,126,147,163,187,196,208,227,240,250,273,293,
%T A382931 314,342,361,384,392,409,434,455,459,483,504,507,525,549,552,579,599,
%U A382931 627,649,679,702,711,718,724,744,752,775,802,829,854,879,894,908,935,960
%N A382931 Numbers k for which the Pythagorean triangle (A046083(k), A046084(k), A009000(k)) has an integer altitude.
%C A382931 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.
%H A382931 Felix Huber, <a href="/A382931/b382931.txt">Table of n, a(n) for n = 1..10000</a>
%H A382931 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PythagoreanTriple.html">Pythagorean Triple</a>
%e A382931 7 is in the sequence because the pythagorean triangle (A046083(7), A046084(7), A009000(7)) = (15, 20, 25) has the integer altitude 15*20/25 = 12.
%p A382931 A382931:=proc(H) # All hypotenuses <= H.
%p A382931     local a,b,c,k,p,q,L,M;
%p A382931     L:=[];
%p A382931     M:=[];
%p A382931     for p from 2 to floor(sqrt(H-1)) do
%p A382931         for q to min(p-1,floor(sqrt(H-p^2))) do
%p A382931             if gcd(p,q)=1 and is(p-q,odd) then
%p A382931                 a:=p^2-q^2;
%p A382931                 b:=2*p*q;
%p A382931                 c:=p^2+q^2;
%p A382931                 for k to iquo(H,c) do
%p A382931                     L:=[op(L),[k*c,k*max(a,b),k*a*b/c]]
%p A382931                 od
%p A382931             fi
%p A382931         od
%p A382931     od;
%p A382931     L:=sort(L);
%p A382931     for k to nops(L) do
%p A382931         if is(L[k,3],integer) then
%p A382931            M:=[op(M),k]
%p A382931         fi
%p A382931     od;
%p A382931     return op(M)
%p A382931 end proc;
%p A382931 A382931(1075);
%Y A382931 Cf. A009000, A046083, A046084, A382932.
%K A382931 nonn
%O A382931 1,1
%A A382931 _Felix Huber_, Apr 11 2025