A377304 a(n) is the number of distinct cuboids whose edges are divisors of n.

1, 4, 4, 10, 4, 20, 4, 20, 10, 20, 4, 56, 4, 20, 20, 35, 4, 56, 4, 56, 20, 20, 4, 120, 10, 20, 20, 56, 4, 120, 4, 56, 20, 20, 20, 165, 4, 20, 20, 120, 4, 120, 4, 56, 56, 20, 4, 220, 10, 56, 20, 56, 4, 120, 20, 120, 20, 20, 4, 364, 4, 20, 56, 84, 20, 120, 4, 56

Offset

1,2

Comments

Equivalently, a(n) is the number of unordered triples of divisors of n.

There are tau(n)*(tau(n) - 1)*(tau(n) - 2)/6 distinct cuboids with three different edges, (tau(n)*1*(tau(n) - 1) + tau(n)*(tau(n) - 1)*2)/3 distinct cuboids with two different edges and tau(n) distinct cuboids that are cubes.

Links

Felix Huber, Table of n, a(n) for n = 1..10000

Formula

a(n) = (tau(n)^3 + 3*tau(n)^2 + 2*tau(n))/6.

a(n) = binomial(tau(n) + 2, 3).

Example

a(4) = 10, because there are 10 distinct cuboids whose edges are divisors of 4: (1, 1, 1), (1, 1, 2), (1, 1, 4), (1, 2, 2), (1, 2, 4), (1, 4, 4), (2, 2, 2), (2, 2, 4), (2, 4, 4), (4, 4, 4).

Maple

A377304:=proc(n)
   local d;
   d:=NumberTheory:-tau(n);
   return (d^3+3*d^2+2*d)/6
end proc;
seq(A377304(n), n=1..68);

Mathematica

a[n_] := Binomial[DivisorSigma[0, n] + 2, 3]; Array[a, 70] (* Amiram Eldar, Nov 07 2024 *)

Crossrefs

Cf. A000005, A086222.

Sequence in context: A286779 A007426 A353267 * A339336 A319056 A369225

Adjacent sequences: A377301 A377302 A377303 * A377305 A377306 A377307

Keyword

nonn

Author

Felix Huber, Oct 25 2024

Status

approved