cp's OEIS Frontend

A369162 a(n) = A000688(A000688(n)).

%I A369162 #9 Jan 15 2024 09:43:05
%S A369162 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,
%T A369162 1,2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,
%U A369162 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1
%N A369162 a(n) = A000688(A000688(n)).
%C A369162 First differs from A364388 at n = 42.
%C A369162 The sums of the first 10^k terms, for k = 1, 2, ..., are 10, 102, 1024, 10285, 102988, 1030280, 10304021, 103043644, 1030448091, 10304515936, ... . 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 1.0304... .
%D A369162 József Sándor, Dragoslav S. Mitrinovic, Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, Chapter XIII, page 477-478.
%H A369162 Amiram Eldar, <a href="/A369162/b369162.txt">Table of n, a(n) for n = 1..10000</a>
%H A369162 Paul Erdős and Aleksandar Ivić, <a href="https://www.jstor.org/stable/44095772">On the iterates of the enumerating function of finite Abelian groups</a>, Bull. Cl. Sci. Math. Nat., Sci. Math., Vol. 17 (1989), pp. 13-22; <a href="https://users.renyi.hu/~p_erdos/1989-23.pdf">alternative link</a>.
%H A369162 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. 4.2.
%F A369162 Sum_{k=1..n} a(k) = c * n + O(sqrt(n) * log(n)^4), where c = Sum_{k>=1} d(k) * A000688(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 A369162 Table[FiniteAbelianGroupCount[FiniteAbelianGroupCount[n]], {n, 1, 100}]
%o A369162 (PARI) A000688(n) = vecprod(apply(numbpart, factor(n)[, 2]));
%o A369162 a(n) = A000688(A000688(n));
%Y A369162 Cf. A000688, A364388.
%Y A369162 Cf. A048109, A059956, A271971.
%Y A369162 Cf. A369163, A369164, A369165, A369167.
%K A369162 nonn,easy
%O A369162 1,36
%A A369162 _Amiram Eldar_, Jan 15 2024