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

%I A377304 #20 Nov 17 2024 07:41:34
%S A377304 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,
%T A377304 20,56,4,120,4,56,20,20,20,165,4,20,20,120,4,120,4,56,56,20,4,220,10,
%U A377304 56,20,56,4,120,20,120,20,20,4,364,4,20,56,84,20,120,4,56
%N A377304 a(n) is the number of distinct cuboids whose edges are divisors of n.
%C A377304 Equivalently, a(n) is the number of unordered triples of divisors of n.
%C A377304 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.
%H A377304 Felix Huber, <a href="/A377304/b377304.txt">Table of n, a(n) for n = 1..10000</a>
%F A377304 a(n) = (tau(n)^3 + 3*tau(n)^2 + 2*tau(n))/6.
%F A377304 a(n) = binomial(tau(n) + 2, 3).
%e A377304 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).
%p A377304 A377304:=proc(n)
%p A377304    local d;
%p A377304    d:=NumberTheory:-tau(n);
%p A377304    return (d^3+3*d^2+2*d)/6
%p A377304 end proc;
%p A377304 seq(A377304(n),n=1..68);
%t A377304 a[n_] := Binomial[DivisorSigma[0, n] + 2, 3]; Array[a, 70] (* _Amiram Eldar_, Nov 07 2024 *)
%Y A377304 Cf. A000005, A086222.
%K A377304 nonn
%O A377304 1,2
%A A377304 _Felix Huber_, Oct 25 2024