A365489 - OEIS

%I #8 Sep 06 2023 01:19:07

%S 1,4,4,4,4,16,4,4,4,16,4,16,4,16,16,7,4,16,4,16,16,16,4,16,4,16,4,16,

%T 4,64,4,7,16,16,16,16,4,16,16,16,4,64,4,16,16,16,4,28,4,16,16,16,4,16,

%U 16,16,16,16,4,64,4,16,16,7,16,64,4,16,16,64,4,16,4

%N The number of divisors of the smallest cube divisible by n.

%C The number of divisors of the cube root of the smallest cube divisible by n, A019555(n), is A365488(n).

%H Amiram Eldar, <a href="/A365489/b365489.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = A000005(A053149(n)).

%F Multiplicative with a(p^e) = 3*ceiling(e/3) + 1.

%F Dirichlet g.f.: zeta(s) * zeta(3*s) * Product_{p prime} (1 + 3/p^s - 1/p^(3*s)).

%t f[p_, e_] := 3*Ceiling[e/3] + 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

%o (PARI) a(n) = vecprod(apply(x -> 3*((x-1)\3) + 4, factor(n)[, 2]));

%Y Cf. A000005, A019555, A053149, A365488.

%K nonn,easy,mult

%O 1,2

%A _Amiram Eldar_, Sep 05 2023