A070103 - OEIS

%I #11 May 14 2019 10:21:45

%S 0,0,0,0,0,0,1,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,1,0,1,0,2,0,1,0,0,0,0,0,

%T 2,0,1,0,3,0,2,0,2,0,1,0,3,0,2,0,1,0,2,0,0,0,0,0,3,0,1,0,4,0,5,0,4,0,

%U 2,0,1,0,1,0,2,0,2,0,3,0,1,0,6,0,4,0,6,0,6,0

%N Number of obtuse integer triangles with perimeter n and prime side lengths.

%H R. Zumkeller, <a href="/A070080/a070080.txt">Integer-sided triangles</a>

%F a(n) = A070093(n) - A070098(n).

%F a(n) = Sum_{k=1..floor(n/3)} Sum_{i=k..floor((n-k)/2)} (1 - sign(floor((i^2 + k^2)/(n-i-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=11 there are A005044(11)=4 integer triangles: [1,5,5], [2,4,5], [3,3,5] and [3,4,4]; only one of the two obtuses ([2,4,5] and [3,3,5]) consists of primes, therefore a(11)=1.

%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[(i^2 + k^2)/(n - i - 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, A070093, A070098, A070101, A070088, A070105, A070108, A070129.

%K nonn

%O 1,27

%A _Reinhard Zumkeller_, May 05 2002