A334134 Number of integer-sided triangles with perimeter n whose side lengths can be written as the sum of two primes in the same number of ways.

0, 0, 1, 0, 1, 1, 2, 1, 1, 0, 0, 1, 1, 2, 3, 3, 4, 5, 6, 6, 6, 5, 5, 4, 6, 4, 6, 5, 5, 6, 7, 6, 7, 6, 5, 6, 4, 4, 7, 5, 3, 6, 6, 7, 7, 9, 6, 8, 5, 6, 8, 7, 5, 6, 7, 5, 7, 5, 6, 4, 5, 3, 8, 4, 6, 6, 8, 7, 9, 9, 10, 7, 9, 8, 12, 6, 8, 7, 9, 6, 11, 6, 11, 9, 11, 6, 14

Offset: 1

Author

Wesley Ivan Hurt, Apr 15 2020

Keywords

• nonn
• easy

Examples

			a(4) = 0; no triangles can be made.
a(7) = 2; The two triangles [1,3,3] and [2,2,3] both have perimeter 7, and in each case, the side lengths can be written as the sum of two primes in the same number of ways (0 ways).
a(12) = 1; The triangle [4,4,4] has perimeter 12 and all of its side lengths can be written as the sum of two primes in the same number of ways (1 way).
a(15) = 3; the triangles [4,4,7], [4,5,6] and [5,5,5] all have perimeter 15. In each triangle, all the side lengths can be written as the sum of two primes in the same number of ways (1 way).
		

Links

• Wikipedia, Integer Triangle

Crossrefs

Cf. A005044, A061358.

Programs

Mathematica

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

Formula

a(n) = Sum_{k=1..floor(n/3)} Sum_{i=k..floor((n-k)/2)} sign(floor((i+k)/(n-i-k+1))) * [c(i) = c(k) = c(n-i-k)], where [] is the Iverson bracket and c = A061358.