%I #31 Jun 16 2020 14:51:02
%S 0,0,0,0,2,2,6,3,3,0,15,10,10,5,33,21,21,14,14,7,7,0,66,55,55,44,135,
%T 111,111,87,87,63,63,39,192,162,162,132,322,273,273,237,237,201,201,
%U 165,441,382,382,323,323,264,264,222,222,180,180,138,573,521,521
%N Sum of the largest side lengths of all integer-sided triangles with perimeter n whose largest side length is prime.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Integer_triangle">Integer Triangle</a>
%F a(n) = Sum_{k=1..floor(n/3)} Sum_{i=k..floor((n-k)/2)} sign(floor((i+k)/(n-i-k+1))) * A010051(n-i-k) * (n-i-k).
%t Table[Sum[Sum[(n - i - k) (PrimePi[n - i - k] - PrimePi[n - i - k - 1]) Sign[Floor[(i + k)/(n - i - k + 1)]], {i, k, Floor[(n - k)/2]}], {k, Floor[n/3]}], {n, 100}]
%Y Cf. A010051, A306426.
%K nonn
%O 1,5
%A _Wesley Ivan Hurt_, Jun 03 2019