A335621 - OEIS

%I #11 Dec 21 2024 02:35:20

%S 0,0,1,0,0,0,0,1,2,1,1,1,0,0,1,1,2,3,4,2,3,2,2,4,5,4,4,3,3,3,3,5,5,4,

%T 4,6,8,5,8,8,12,7,12,7,10,6,10,10,15,14,20,14,18,17,21,20,25,20,23,18,

%U 19,16,20,22,24,21,25,21,22,20,22,23,28,22,28,22,24,20,23,25

%N Number of integer-sided triangles with perimeter n such that the sum of each pair of side lengths is squarefree.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Integer_triangle">Integer Triangle</a>

%F a(n) = Sum_{k=1..floor(n/3)} Sum_{i=k..floor((n-k)/2)} sign(floor((i+k)/(n-i-k+1))) * mu(i+k)^2 * mu(n-i)^2 * mu(n-k)^2, where mu is the Möbius function (A008683).

%t Table[Sum[Sum[MoebiusMu[i + k]^2*MoebiusMu[n - i]^2*MoebiusMu[n - k]^2 * Sign[Floor[(i + k)/(n - i - k + 1)]], {i, k, Floor[(n - k)/2]}], {k, Floor[n/3]}], {n, 100}]

%Y Cf. A008683, A308061 (each side length is squarefree).

%Y Cf. A335621 (averages of each pair of side lengths is prime).

%K nonn

%O 1,9

%A _Wesley Ivan Hurt_, Oct 02 2020