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A381336 a(n) is the smallest k > 0 for which a nondegenerate integer-sided triangle (k, k + n, c >= k + n) with an integer area exists.

%I A381336 #14 Mar 24 2025 06:14:18
%S A381336 3,6,9,12,12,18,5,7,4,24,14,36,15,10,36,14,7,8,6,21,8,3,12,5,10,15,12,
%T A381336 20,46,35,9,28,20,14,25,16,15,12,22,21,19,16,12,6,20,5,4,10,11,20,21,
%U A381336 30,96,24,13,9,18,7,25,63,21,18,22,9,35,9,25,21,36,17,13
%N A381336 a(n) is the smallest k > 0 for which a nondegenerate integer-sided triangle (k, k + n, c >= k + n) with an integer area exists.
%C A381336 Longest sides c are in A381337.
%H A381336 Felix Huber, <a href="/A381336/b381336.txt">Table of n, a(n) for n = 1..10000</a>
%H A381336 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/HeronsFormula.html">Heron's Formula</a>.
%e A381336 a(5) = 12 because the nondegenerate integer-sided triangle (12, 12 + 5, 25 >= 12 + 5) has an integer area (90), and there is no smaller k > 0 than 12 that satisfies this condition.
%p A381336 A381336:=proc(n)
%p A381336     local k,c,s;
%p A381336     for k do
%p A381336         for c from k+n to 2*k+n-1 do
%p A381336             s:=(n+2*k+c)/2;
%p A381336             if issqr(s*(s-k)*(s-k-n)*(s-c)) then
%p A381336                 return k
%p A381336             fi
%p A381336         od
%p A381336     od;
%p A381336 end proc;
%p A381336 seq(A381336(n),n=1..71);
%Y A381336 Cf. A103974, A103975, A188158, A379830, A381337.
%K A381336 nonn
%O A381336 1,1
%A A381336 _Felix Huber_, Mar 16 2025