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

1,6

Links

Table of n, a(n) for n=1..70.

Wikipedia, Integer Triangle

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.

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}]

Crossrefs

Cf. A010051, A070092.

Sequence in context: A198124 A120207 A202703 * A308165 A090928 A199174

Adjacent sequences: A308144 A308145 A308146 * A308148 A308149 A308150

Keyword

nonn

Author

Wesley Ivan Hurt, May 14 2019

Status

approved