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