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A382488 The number of unitary 3-smooth divisors of n.

%I A382488 #7 Mar 29 2025 04:24:09
%S A382488 1,2,2,2,1,4,1,2,2,2,1,4,1,2,2,2,1,4,1,2,2,2,1,4,1,2,2,2,1,4,1,2,2,2,
%T A382488 1,4,1,2,2,2,1,4,1,2,2,2,1,4,1,2,2,2,1,4,1,2,2,2,1,4,1,2,2,2,1,4,1,2,
%U A382488 2,2,1,4,1,2,2,2,1,4,1,2,2,2,1,4,1,2,2
%N A382488 The number of unitary 3-smooth divisors of n.
%C A382488 Period 6: repeat [1, 2, 2, 2, 1, 4].
%C A382488 Decimal expansion of 407380/333333.
%C A382488 Continued fraction expansion of 10/(6 + sqrt(66)) (with offset 0).
%H A382488 Amiram Eldar, <a href="/A382488/b382488.txt">Table of n, a(n) for n = 1..1000</a>
%H A382488 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,0,0,1).
%F A382488 Multiplicative with a(p^e) = 2 if p <= 3, and 1 otherwise.
%F A382488 a(n) = A034444(A065331(n)).
%F A382488 a(n) = A034444(n) if and only if n is 3-smooth (A003586).
%F A382488 a(n) = A072078(n) if and only if n is squarefree (A005117).
%F A382488 a(n) = abs(A181982(n+9)).
%F A382488 Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2.
%F A382488 G.f.: -(4*x^6 + x^5 + 2*x^4 + 2*x^3 +2*x^2 + x)/(x^6 - 1).
%F A382488 Dirichlet g.f.: (1 + 1/2^s) * (1 + 1/3^s) * zeta(s).
%t A382488 Table[{1, 2, 2, 2, 1, 4}, {12}] // Flatten
%o A382488 (PARI) a(n) = [1, 2, 2, 2, 1, 4][(n-1) % 6 + 1];
%Y A382488 Cf. A005117, A003586, A034444, A065331, A072078, A181982, A382487.
%Y A382488 The number of unitary prime(k)-smooth divisors of n: A134451 (k = 1), this sequence (k = 2), A382489 (k = 3).
%K A382488 nonn,easy,mult
%O A382488 1,2
%A A382488 _Amiram Eldar_, Mar 29 2025