A335621 Number of integer-sided triangles with perimeter n such that the sum of each pair of side lengths is squarefree.

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

Offset

1,9

Links

Table of n, a(n) for n=1..80.

Wikipedia, Integer Triangle

Formula

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).

Mathematica

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}]

Crossrefs

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

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

Sequence in context: A086831 A191340 A211229 * A316675 A111405 A089053

Adjacent sequences: A335618 A335619 A335620 * A335622 A335623 A335624

Keyword

nonn

Author

Wesley Ivan Hurt, Oct 02 2020

Status

approved