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%I #6 Jun 18 2020 14:00:32
%S 0,0,0,0,2,2,5,3,3,0,12,9,9,5,27,18,18,13,13,7,7,0,51,45,45,38,108,93,
%T 93,76,76,57,57,36,153,133,133,111,256,222,222,199,199,174,174,147,
%U 357,316,316,272,272,225,225,193,193,159,159,123,453,420,420
%N Take all the integer-sided triangles with perimeter n and side lengths a, b and c such that a <= b <= c and c 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(n-i-k) * i.
%t Table[Sum[Sum[i (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
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