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

Author

Felix Huber, Oct 25 2024

Keywords

• nonn

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.

Examples

			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).
		

Links

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

Crossrefs

Cf. A000005, A086222.

Programs

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 *)

Formula

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

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