%I #9 Aug 24 2020 14:32:53
%S 0,0,0,0,0,6,7,8,9,0,11,12,13,0,15,16,34,0,19,0,21,0,0,24,50,0,54,28,
%T 58,0,31,0,66,0,35,36,74,0,117,40,123,0,86,0,135,0,47,48,147,0,153,0,
%U 106,0,55,0,171,0,59,60,122,0,189,64,260,0,134,0,276,0
%N Sum of the perimeters of all integer-sided isosceles triangles with perimeter n and prime side lengths.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Integer_triangle">Integer Triangle</a>
%F a(n) = n * Sum_{k=1..floor(n/3)} Sum_{i=k..floor((n-k)/2)} sign(floor((i+k)/(n-i-k+1))) * ([i = k] + [i = n-i-k] - [k = n-i-k]) * c(i) * c(k) * c(n-i-k), where c is the prime characteristic (A010051) and [] is the Iverson bracket.
%t Table[n*Sum[Sum[(PrimePi[i] - PrimePi[i - 1]) (PrimePi[k] - PrimePi[k - 1]) (PrimePi[n - i - k] - PrimePi[n - i - k - 1]) (KroneckerDelta[i, k] + KroneckerDelta[i, n - i - k] - KroneckerDelta[k, n - i - k]) Sign[Floor[(i + k)/(n - i - k + 1)]], {i, k, Floor[(n - k)/2]}], {k, Floor[n/3]}], {n, 100}]
%Y Cf. A010051, A070092.
%K nonn
%O 1,6
%A _Wesley Ivan Hurt_, May 14 2019