A303222 Total volume of all rectangular prisms with dimensions p, q and (p + q)/2 such that p and q are squarefree, n = p + q and p <= q.

0, 1, 3, 14, 15, 42, 56, 136, 144, 230, 220, 612, 611, 665, 675, 1576, 1768, 1836, 1729, 4200, 3528, 4279, 3404, 7524, 6625, 8333, 8289, 14336, 11165, 12675, 10323, 20592, 17688, 23307, 17570, 40410, 27861, 30153, 28899, 52180, 42804, 45864, 55169, 84920

Offset: 1

Author

Wesley Ivan Hurt, Apr 19 2018

Keywords

• nonn
• easy

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• Index entries for sequences related to partitions

Crossrefs

Cf. A008683, A262351.

Programs

Magma

[0] cat [&+[k*(n-k)*(MoebiusMu(k)^2*MoebiusMu(n-k)^2)*n/2: k in [1..Floor(n/2)]]: n in [2..60]]; // Vincenzo Librandi, Apr 21 2018

Maple

N:= 100: # for a(1)..a(N)
A:= Vector(N):
SF:= select(numtheory:-issqrfree, [$1..N-1]):
for iq from 1 to nops(SF) do
  q:= SF[iq];
  for ip from 1 to iq do
    p:= SF[ip];
    n:= p+q;
    if n > N then break fi;
    A[n]:= A[n] + p*q*(p+q)/2
  od
od:
convert(A,list); # Robert Israel, Jun 12 2018

Mathematica

Table[(n/2)*Sum[i (n - i)*MoebiusMu[i]^2 MoebiusMu[n - i]^2, {i, Floor[n/2]}], {n, 80}]

PARI

a(n) = n*sum(i=1, n\2, i*(n-i)*moebius(i)^2*moebius(n-i)^2)/2; \\ Michel Marcus, Apr 21 2018

Formula

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