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%I #8 Nov 27 2020 17:39:00
%S 0,0,0,0,0,2,2,3,6,7,11,9,14,11,22,18,31,26,41,30,54,41,68,53,83,66,
%T 99,73,109,80,119,87,140,105,162,124,185,144,222,178,261,214,302,241,
%U 334,269,367,298,401,328,453,363,494,399,536,436,598,493,662,552
%N Take all the integer-sided triangles with perimeter n and side lengths a, b and c such that a <= b <= c and a is prime. a(n) is the sum of all the b's.
%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(k) * i.
%t Table[Sum[Sum[i (PrimePi[k] - PrimePi[k - 1]) Sign[Floor[(i + k)/(n - i - k + 1)]], {i, k, Floor[(n - k)/2]}], {k, Floor[n/3]}], {n, 100}]
%t Table[Total[Select[IntegerPartitions[n,{3}],PrimeQ[#[[3]]]&&#[[3]]+#[[2]]>#[[1]]&][[All,2]]],{n,100}] (* _Harvey P. Dale_, Nov 27 2020 *)
%Y Cf. A010051, A308450, A308513.
%K nonn
%O 1,6
%A _Wesley Ivan Hurt_, Jun 02 2019
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