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

Author

Wesley Ivan Hurt, May 14 2019

Keywords

• nonn
• look

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• Wikipedia, Integer Triangle

Crossrefs

Cf. A008683, A308061, A308116.

Programs

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}]

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

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).