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%I #22 Sep 19 2023 01:43:52
%S 1,9,9,9,9,81,9,9,9,81,9,81,9,81,81,9,9,81,9,81,81,81,9,81,9,81,9,81,
%T 9,729,9,9,81,81,81,81,9,81,81,81,9,729,9,81,81,81,9,81,9,81,81,81,9,
%U 81,81,81,81,81,9,729,9,81,81,9,81,729,9,81,81,729,9,81,9,81,81,81
%N a(n) = 9^omega(n), where omega(n) is the number of distinct primes dividing n.
%F a(n) = A001019(A001221(n)).
%F Multiplicative with a(p^e) = 9.
%F a(n) = Sum_{d|n} mu(d)^2 * tau(d)^3.
%F Dirichlet g.f.: Product_{p prime} (1 + 9/(p^s-1)). - _Amiram Eldar_, Sep 19 2023
%t Table[9^PrimeNu[n], {n, 1, 100}] (* _Amiram Eldar_, May 15 2021 *)
%o (PARI) a(n) = 9^omega(n);
%o (PARI) a(n) = prod(k=1, #f=factor(n)[, 2], 9);
%o (PARI) a(n) = sumdiv(n, d, moebius(d)^2*numdiv(d)^3);
%Y k^omega(n): A034444 (k=2), A074816 (k=3), A082476 (k=5), this sequence (k=9).
%Y Cf. A000005, A001019, A001221.
%K nonn,mult
%O 1,2
%A _Seiichi Manyama_, May 15 2021
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