A369163 - OEIS

%I #9 Jan 15 2024 09:48:19

%S 1,1,1,2,1,1,1,2,2,1,1,2,1,1,1,2,1,2,1,2,1,1,1,2,2,1,2,2,1,1,1,2,1,1,

%T 1,3,1,1,1,2,1,1,1,2,2,1,1,2,2,2,1,2,1,2,1,2,1,1,1,2,1,1,2,2,1,1,1,2,

%U 1,1,1,4,1,1,2,2,1,1,1,2,2,1,1,2,1,1,1

%N a(n) = A000005(A000688(n)).

%C First differs from A007424, A278908, A307848, A323308, A358260 and A365549 at n = 36.

%C The sums of the first 10^k terms, for k = 1, 2, ..., are 13, 143, 1486, 15054, 151067, 1511982, 15123465, 151245456, 1512484372, 15124927227, ... . 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 1.512... .

%D József Sándor, Dragoslav S. Mitrinovic, Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, Chapter II, page 73.

%H Amiram Eldar, <a href="/A369163/b369163.txt">Table of n, a(n) for n = 1..10000</a>

%H 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. 4.3.

%F Sum_{k=1..n} a(k) = c * n + O(sqrt(n) * log(n)^4), where c = Sum_{k>=1} d(k) * A000005(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 Table[DivisorSigma[0, FiniteAbelianGroupCount[n]], {n, 1, 100}]

%o (PARI) a(n) = numdiv(vecprod(apply(numbpart, factor(n)[, 2])));

%Y Cf. A000005, A000688.

%Y Cf. A048109, A059956, A271971.

%Y Cf. A369162, A369164, A369165.

%Y Cf. A007424, A278908, A307848, A323308, A358260, A365549.

%K nonn,easy

%O 1,4

%A _Amiram Eldar_, Jan 15 2024