A308140 Sum of the largest side lengths of all integer-sided triangles with squarefree side lengths and perimeter n.

0, 0, 1, 0, 2, 2, 6, 3, 3, 0, 10, 5, 17, 12, 32, 20, 20, 13, 14, 7, 27, 30, 64, 43, 32, 21, 71, 48, 92, 92, 154, 112, 110, 85, 169, 123, 142, 94, 222, 154, 171, 101, 245, 169, 316, 250, 424, 321, 361, 263, 322, 219, 367, 337, 348, 260, 275, 242, 405, 310

Offset

1,5

Links

Robert Israel, Table of n, a(n) for n = 1..10000

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)^2 * mu(k)^2 * mu(n-i-k)^2 * (n-i-k), where mu is the Möbius function (A008683).

Maple

N:= 100: # for a(1)..a(N)
SF:= select(numtheory:-issqrfree, [$1..N/2]):
V:= Vector(N):
for ia from 1 to nops(SF) do
    a:= SF[ia];
    if 2*a >= N then break fi;
    for ib from ia by -1 to 1 do
      b:= SF[ib];
      if 2*b <= a then break fi;
      cs:= select(c -> b+c > a, SF[1...ib]);
      P:= select(`<=`, map(c -> a+b+c, cs), N);
      V[P]:= V[P] +~ a;
od od:
convert(V, list); # Robert Israel, May 14 2019

Mathematica

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

Prog

(PARI) a(n) = sum(k=1, n\3, sum(i=k, (n-k)\2, sign((i+k)\(n-i-k+1))* issquarefree(i)*issquarefree(k)*issquarefree(n-i-k)*(n-i-k))); \\ Michel Marcus, May 14 2019

Crossrefs

Cf. A008683, A308061, A308116.

Sequence in context: A355076 A344007 A130478 * A306464 A338449 A163890

Adjacent sequences: A308137 A308138 A308139 * A308141 A308142 A308143

Keyword

nonn,look

Author

Wesley Ivan Hurt, May 14 2019

Status

approved