A308147 Sum of the perimeters of all integer-sided isosceles triangles with perimeter n and prime side lengths.

0, 0, 0, 0, 0, 6, 7, 8, 9, 0, 11, 12, 13, 0, 15, 16, 34, 0, 19, 0, 21, 0, 0, 24, 50, 0, 54, 28, 58, 0, 31, 0, 66, 0, 35, 36, 74, 0, 117, 40, 123, 0, 86, 0, 135, 0, 47, 48, 147, 0, 153, 0, 106, 0, 55, 0, 171, 0, 59, 60, 122, 0, 189, 64, 260, 0, 134, 0, 276, 0

Offset: 1

Author

Wesley Ivan Hurt, May 14 2019

Keywords

• nonn

Links

• Wikipedia, Integer Triangle

Crossrefs

Cf. A010051, A070092.

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]) (KroneckerDelta[i, k] + KroneckerDelta[i, n - i - k] - KroneckerDelta[k, n - i - k]) Sign[Floor[(i + k)/(n - i - k + 1)]], {i, k, Floor[(n - k)/2]}], {k, Floor[n/3]}], {n, 100}]

Formula

a(n) = n * Sum_{k=1..floor(n/3)} Sum_{i=k..floor((n-k)/2)} sign(floor((i+k)/(n-i-k+1))) * ([i = k] + [i = n-i-k] - [k = n-i-k]) * c(i) * c(k) * c(n-i-k), where c is the prime characteristic (A010051) and [] is the Iverson bracket.