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A369164 a(n) = A001221(A000688(n)).

%I A369164 #7 Jan 15 2024 09:48:47
%S A369164 0,0,0,1,0,0,0,1,1,0,0,1,0,0,0,1,0,1,0,1,0,0,0,1,1,0,1,1,0,0,0,1,0,0,
%T A369164 0,1,0,0,0,1,0,0,0,1,1,0,0,1,1,1,0,1,0,1,0,1,0,0,0,1,0,0,1,1,0,0,0,1,
%U A369164 0,0,0,2,0,0,1,1,0,0,0,1,1,0,0,1,0,0,0
%N A369164 a(n) = A001221(A000688(n)).
%C A369164 First differs from A369165 at n = 36, from A080733 at n = 49, and from A107078 at n = 72.
%C A369164 The sums of the first 10^k terms, for k = 1, 2, ..., are 3, 40, 426, 4307, 43203, 432211, 4322486, 43226028, 432261887, 4322622387, ... . From these values the asymptotic mean of this sequence, whose existence was proven by Ivić (1983) (see the Formula section), can be empirically evaluated by 0.43226... .
%D A369164 József Sándor, Dragoslav S. Mitrinovic, Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, Chapter V, page 164.
%H A369164 Amiram Eldar, <a href="/A369164/b369164.txt">Table of n, a(n) for n = 1..10000</a>
%H A369164 Aleksandar Ivić, <a href="https://doi.org/10.1016/0022-314X(83)90037-9">On the number of abelian groups of a given order and on certain related multiplicative functions</a>, Journal of Number Theory, Vol. 16, No. 1 (1983), pp. 119-137. See p. 131, eq. 4.4.
%F A369164 Sum_{k=1..n} a(k) = c * n + O(sqrt(n) * log(n)^3/log(log(n))^2), where c = Sum_{k>=1} d(k) * A001221(k) is a constant, d(k) is the asymptotic density of the set {m | A000688(m) = k} (e.g., d(1) = A059956, d(2) = A271971, d(3) appears in A048109) (Ivić, 1983).
%t A369164 Table[PrimeNu[FiniteAbelianGroupCount[n]], {n, 1, 100}]
%o A369164 (PARI) a(n) = omega(vecprod(apply(numbpart, factor(n)[, 2])));
%Y A369164 Cf. A001221, A000688, A080733, A107078.
%Y A369164 Cf. A048109, A059956, A271971.
%Y A369164 Cf. A369162, A369163, A369165.
%K A369164 nonn,easy
%O A369164 1,72
%A A369164 _Amiram Eldar_, Jan 15 2024