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A360540 a(n) is the cubefull part of n: the largest divisor of n that is a cubefull number (A036966).

%I A360540 #13 Feb 12 2023 17:27:47
%S A360540 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 A360540 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 A360540 1,1,1,1,1,8,1,1,1,1,1,1,1,16,81,1,1,1,1
%N A360540 a(n) is the cubefull part of n: the largest divisor of n that is a cubefull number (A036966).
%H A360540 Amiram Eldar, <a href="/A360540/b360540.txt">Table of n, a(n) for n = 1..10000</a>
%F A360540 a(n) = 1 if and only if n is a cubefree number (A004709).
%F A360540 a(n) = n if and only if n is a cubefull number (A036966).
%F A360540 a(n) <= A057521(n) with equality if and only if n is in A337050.
%F A360540 a(n) = n/A360539(n).
%F A360540 Multiplicative with a(p^e) = p^e if e >= 3, and 1 otherwise.
%F A360540 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 A360540 f[p_, e_] := If[e > 2, p^e, 1]; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
%o A360540 (PARI) a(n) = {my(f = factor(n)); prod(i=1, #f~, if(f[i, 2] > 2, f[i, 1]^f[i, 2], 1));}
%Y A360540 Cf. A004709, A036966, A057521, A360539.
%K A360540 nonn,easy,mult
%O A360540 1,8
%A A360540 _Amiram Eldar_, Feb 11 2023