A308165 Sum of the perimeters of all integer-sided triangles with perimeter n and prime sides.

0, 0, 0, 0, 0, 6, 7, 8, 9, 0, 11, 12, 13, 0, 30, 16, 34, 0, 19, 0, 21, 0, 23, 24, 50, 0, 81, 28, 87, 0, 62, 0, 66, 0, 105, 36, 111, 0, 195, 40, 205, 0, 172, 0, 135, 0, 235, 48, 245, 0, 204, 0, 212, 0, 110, 0, 171, 0, 295, 60, 183, 0, 378, 64, 520, 0, 335, 0

Offset: 1

Author

Wesley Ivan Hurt, May 15 2019

Keywords

• nonn

Links

• Wikipedia, Integer Triangle

Crossrefs

Cf. A070088.

Programs

Mathematica

Table[n*Sum[Sum[(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}]

Formula

a(n) = n * A070088(n).

a(n) = 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).