%I #13 Feb 12 2023 17:27:47

%S 1,1,1,1,1,1,1,8,1,1,1,1,1,1,1,16,1,1,1,1,1,1,1,8,1,1,27,1,1,1,1,32,1,

%T 1,1,1,1,1,1,8,1,1,1,1,1,1,1,16,1,1,1,1,1,27,1,8,1,1,1,1,1,1,1,64,1,1,

%U 1,1,1,1,1,8,1,1,1,1,1,1,1,16,81,1,1,1,1

%N a(n) is the cubefull part of n: the largest divisor of n that is a cubefull number (A036966).

%H Amiram Eldar, <a href="/A360540/b360540.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = 1 if and only if n is a cubefree number (A004709).

%F a(n) = n if and only if n is a cubefull number (A036966).

%F a(n) <= A057521(n) with equality if and only if n is in A337050.

%F a(n) = n/A360539(n).

%F Multiplicative with a(p^e) = p^e if e >= 3, and 1 otherwise.

%F Dirichlet g.f.: zeta(s-1) * Product_{p prime} (1 - p^(1-s) + p^(-s) - p^(1-3*s) - p^(1-2*s) + p^(-2*s) + p^(3-3*s)).

%t f[p_, e_] := If[e > 2, p^e, 1]; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

%o (PARI) a(n) = {my(f = factor(n)); prod(i=1, #f~, if(f[i, 2] > 2, f[i, 1]^f[i, 2], 1));}

%Y Cf. A004709, A036966, A057521, A360539.

%K nonn,easy,mult

%O 1,8

%A _Amiram Eldar_, Feb 11 2023