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Number of integer-sided triangles with perimeter n such that the average of each pair of side lengths is prime.
1

%I #10 Feb 01 2021 21:21:56

%S 0,0,0,0,0,1,1,0,1,0,1,0,0,0,1,0,1,0,1,0,1,0,0,0,1,0,1,0,0,0,0,0,1,0,

%T 1,0,1,0,2,0,2,0,2,0,1,0,1,0,2,0,2,0,1,0,1,0,2,0,1,0,1,0,2,0,3,0,2,0,

%U 2,0,1,0,1,0,2,0,1,0,0,0,1,0,2,0,1,0,2,0,3,0,2,0,1,0,1,0,1,0,2,0

%N Number of integer-sided triangles with perimeter n such that the average of each pair of side lengths 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))) * chi((i+k)/2) * chi((i+k)/2) * chi((n-k)/2) * c((i+k)/2) * c((i+k)/2) * c((n-k)/2), where c is the prime characteristic (A010051) and chi(n) = 1 - ceiling(n) + floor(n).

%t Table[Sum[Sum[(1 - Ceiling[(i + k)/2] + Floor[(i + k)/2]) (1 - Ceiling[(n - i)/2] + Floor[(n - i)/2]) (1 - Ceiling[(n - k)/2] + Floor[(n - k)/2]) (PrimePi[(i + k)/2] - PrimePi[(i + k)/2 - 1])*(PrimePi[(n - i)/2] - PrimePi[(n - i)/2 - 1])*(PrimePi[(n - k)/2] - PrimePi[(n - k)/2 - 1]) * Sign[Floor[(i + k)/(n - i - k + 1)]], {i, k, Floor[(n - k)/2]}], {k, Floor[n/3]}], {n, 100}]

%Y Cf. A010051, A335623.

%K nonn

%O 1,39

%A _Wesley Ivan Hurt_, Oct 02 2020