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%I #8 Oct 01 2023 17:21:40
%S 1,1,1,1,1,1,1,4,1,1,1,1,1,1,1,5,1,1,1,1,1,1,1,4,1,1,4,1,1,1,1,6,1,1,
%T 1,1,1,1,1,4,1,1,1,1,1,1,1,5,1,1,1,1,1,4,1,4,1,1,1,1,1,1,1,7,1,1,1,1,
%U 1,1,1,4,1,1,1,1,1,1,1,5,5,1,1,1,1,1,1
%N The number of divisors of the largest divisor of n that is a cubefull number (A036966).
%H Amiram Eldar, <a href="/A366145/b366145.txt">Table of n, a(n) for n = 1..10000</a>
%F a(n) = A000005(A360540(n)).
%F a(n) = A000005(n)/A366147(n).
%F a(n) >= 1, with equality if and only if n is cubefree (A004709).
%F a(n) <= A000005(n), with equality if and only if n is cubefull (A036966).
%F Multiplicative with a(p^e) = 1 if e <= 2 and e+1 otherwise.
%F Dirichlet g.f.: zeta(s)^2 * Product_{p prime} (1 - 1/p^s + 3/p^(3*s) - 2/p^(4*s)).
%F Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = zeta(2) * Product_{p prime} (1 - 1/p^2 + 3/p^3 + 1/p^4 - 2/p^5) = 1.76434793373691907811... .
%t f[p_, e_] := If[e < 3, 1, e+1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
%o (PARI) a(n) = vecprod(apply(x -> if(x < 3, 1, x+1), factor(n)[, 2]));
%Y Cf. A000005, A036966, A357669, A360540, A366076, A366146, A366147.
%K nonn,easy,mult
%O 1,8
%A _Amiram Eldar_, Oct 01 2023
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