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%I #6 Jun 18 2020 13:39:15
%S 0,0,0,0,0,2,2,3,3,0,3,5,5,0,10,7,12,0,7,0,7,0,7,11,18,0,29,13,35,0,
%T 24,0,24,0,35,17,43,0,71,19,77,0,62,0,49,0,83,23,91,0,76,0,76,0,38,0,
%U 59,0,101,29,67,0,140,31,194,0,121,0,125,0,104,0,83
%N Take all the integer-sided triangles with perimeter n and prime sides a, b, and c such that a <= b <= c. 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(i) * A010051(k) * A010051(n-i-k) * i.
%t Table[Sum[Sum[i (PrimePi[i] - PrimePi[i - 1]) (PrimePi[k] - PrimePi[k - 1]) (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. A308109.
%K nonn
%O 1,6
%A _Wesley Ivan Hurt_, May 13 2019
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