A070095 Number of acute integer triangles with perimeter n and prime side lengths.

0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 1, 2, 0, 1, 0, 1, 0, 0, 1, 1, 0, 1, 1, 2, 0, 2, 0, 2, 0, 1, 1, 2, 0, 2, 1, 3, 0, 2, 0, 2, 0, 2, 1, 3, 0, 3, 0, 2, 0, 2, 0, 3, 0, 2, 1, 2, 0, 2, 1, 3, 0, 1, 0, 3, 0, 3, 0, 2, 0, 3, 1, 4, 0, 3, 0, 3, 0, 1, 1, 3, 0, 3, 1, 4, 0

Offset: 1

Author

Reinhard Zumkeller, May 05 2002

Keywords

• nonn

Examples

			For n=17 there are A005044(17)=8 integer triangles: [1,8,8], [2,7,8], [3,6,8], [3,7,7], [4,5,8], [4,6,7], [5,5,7] and [5,6,6]: the two consisting of primes ([3,7,7] and [5,5,7]) are also acute, therefore a(17)=2.
		

Links

• R. Zumkeller, Integer-sided triangles

Crossrefs

Cf. A070080, A070081, A070082, A070088, A070093, A070097, A070100, A070103, A070120.

Programs

Mathematica

Table[Sum[Sum[(PrimePi[i] - PrimePi[i - 1]) (PrimePi[k] - PrimePi[k - 1]) (PrimePi[n - i - k] - PrimePi[n - i - k - 1]) (1 - Sign[Floor[(n - i - k)^2/(i^2 + k^2)]]) Sign[Floor[(i + k)/(n - i - k + 1)]], {i, k, Floor[(n - k)/2]}], {k, Floor[n/3]}], {n, 100}] (* Wesley Ivan Hurt, May 13 2019 *)

Formula

a(n) = A070088(n) - A070103(n).

a(n) = Sum_{k=1..floor(n/3)} Sum_{i=k..floor((n-k)/2)} (1 - sign(floor((n-i-k)^2/(i^2+k^2)))) * sign(floor((i+k)/(n-i-k+1))) * A010051(i) * A010051(k) * A010051(n-i-k). - Wesley Ivan Hurt, May 13 2019