A308318 Number of integer-sided triangles with perimeter n and at least one nonprime side length.

0, 0, 1, 0, 1, 0, 1, 0, 2, 2, 3, 2, 4, 4, 5, 4, 6, 7, 9, 8, 11, 10, 13, 11, 14, 14, 16, 15, 18, 19, 22, 21, 25, 24, 27, 26, 30, 30, 32, 32, 35, 37, 40, 40, 45, 44, 47, 47, 51, 52, 57, 56, 61, 61, 68, 65, 72, 70, 75, 74, 82, 80, 85, 84, 88, 91, 97, 96, 103

Offset: 1

Author

Wesley Ivan Hurt, May 19 2019

Keywords

• nonn

Links

• Wikipedia, Integer Triangle

Crossrefs

Cf. A005044, A010051, A070088.

Programs

Mathematica

Table[Sum[Sum[(1 - (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) = Sum_{k=1..floor(n/3)} Sum_{i=k..floor((n-k)/2)} sign(floor((i+k)/(n-i-k+1))) * (1 - A010051(i) * A010051(k) * A010051(n-i-k)).

a(n) = A005044(n) - A070088(n).