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A369165 a(n) = A001222(A000688(n)).

%I A369165 #13 Apr 26 2025 20:53:58
%S A369165 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 A369165 0,2,0,0,0,1,0,0,0,1,1,0,0,1,1,1,0,1,0,1,0,1,0,0,0,1,0,0,1,1,0,0,0,1,
%U A369165 0,0,0,2,0,0,1,1,0,0,0,1,1,0,0,1,0,0,0
%N A369165 a(n) = A001222(A000688(n)).
%C A369165 First differs from A369164 at n = 36.
%C A369165 The sums of the first 10^k terms, for k = 1, 2, ..., are 3, 42, 450, 4592, 46185, 462402, 4625478, 46258861, 462599818, 4626029362, ... . 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 0.4626... .
%C A369165 First differs from A056170 at n=128, 256, 384, 512, 640.... - _R. J. Mathar_, Jan 18 2024
%D A369165 József Sándor, Dragoslav S. Mitrinovic, Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, Chapter V, page 164.
%H A369165 Amiram Eldar, <a href="/A369165/b369165.txt">Table of n, a(n) for n = 1..10000</a>
%H A369165 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. 4.5.
%F A369165 Sum_{k=1..n} a(k) = c * n + O(sqrt(n) * log(n)^3/log(log(n))), where c = Sum_{k>=1} d(k) * A001222(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 A369165 Table[PrimeOmega[FiniteAbelianGroupCount[n]], {n, 1, 100}]
%o A369165 (PARI) a(n) = bigomega(vecprod(apply(numbpart, factor(n)[, 2])));
%Y A369165 Cf. A001222, A000688.
%Y A369165 Cf. A048109, A059956, A271971.
%Y A369165 Cf. A369162, A369163, A369164.
%K A369165 nonn,easy
%O A369165 1,36
%A A369165 _Amiram Eldar_, Jan 15 2024