%I #11 May 14 2019 10:22:36
%S 0,0,0,0,0,1,0,1,1,0,0,1,1,0,1,1,2,0,1,0,1,0,0,1,1,0,1,1,2,0,2,0,2,0,
%T 1,1,2,0,2,1,3,0,2,0,2,0,2,1,3,0,3,0,2,0,2,0,3,0,2,1,2,0,2,1,3,0,1,0,
%U 3,0,3,0,2,0,3,1,4,0,3,0,3,0,1,1,3,0,3,1,4,0
%N Number of acute integer triangles with perimeter n and prime side lengths.
%H R. Zumkeller, <a href="/A070080/a070080.txt">Integer-sided triangles</a>
%F a(n) = A070088(n) - A070103(n).
%F a(n) = Sum_{k=1..floor(n/3)} Sum_{i=k..floor((n-k)/2)} (1 - sign(floor((n-i-k)^2/(i^2+k^2)))) * sign(floor((i+k)/(n-i-k+1))) * A010051(i) * A010051(k) * A010051(n-i-k). - _Wesley Ivan Hurt_, May 13 2019
%e For n=17 there are A005044(17)=8 integer triangles: [1,8,8], [2,7,8], [3,6,8], [3,7,7], [4,5,8], [4,6,7], [5,5,7] and [5,6,6]: the two consisting of primes ([3,7,7] and [5,5,7]) are also acute, therefore a(17)=2.
%t Table[Sum[Sum[(PrimePi[i] - PrimePi[i - 1]) (PrimePi[k] - PrimePi[k - 1]) (PrimePi[n - i - k] - PrimePi[n - i - k - 1]) (1 - Sign[Floor[(n - i - k)^2/(i^2 + k^2)]]) Sign[Floor[(i + k)/(n - i - k + 1)]], {i, k, Floor[(n - k)/2]}], {k, Floor[n/3]}], {n, 100}] (* _Wesley Ivan Hurt_, May 13 2019 *)
%Y Cf. A070080, A070081, A070082, A070088, A070093, A070097, A070100, A070103, A070120.
%K nonn
%O 1,17
%A _Reinhard Zumkeller_, May 05 2002