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A058060 Number of distinct prime factors of d(n), the number of divisors of n.

%I A058060 #27 Jan 15 2024 01:45:26
%S A058060 0,1,1,1,1,1,1,1,1,1,1,2,1,1,1,1,1,2,1,2,1,1,1,1,1,1,1,2,1,1,1,2,1,1,
%T A058060 1,1,1,1,1,1,1,1,1,2,2,1,1,2,1,2,1,2,1,1,1,1,1,1,1,2,1,1,2,1,1,1,1,2,
%U A058060 1,1,1,2,1,1,2,2,1,1,1,2,1,1,1,2,1,1,1,1,1,2,1,2,1,1,1,2,1,2,2,1,1,1,1,1,1
%N A058060 Number of distinct prime factors of d(n), the number of divisors of n.
%C A058060 The sums of the first 10^k terms, for k = 1, 2, ..., are 9, 122, 1285, 13096, 131729, 1319621, 13203252, 132055132, 1320621032, 13206429426, 132064984784, ... . From these values the asymptotic mean of this sequence, whose existence was proven by Rieger (1972) and Heppner (1974) (see the Formula section), can be empirically evaluated by 1.3206... . - _Amiram Eldar_, Jan 15 2024
%D A058060 József Sándor, Dragoslav S. Mitrinovic, Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, Chapter V, page 164.
%H A058060 G. C. Greubel, <a href="/A058060/b058060.txt">Table of n, a(n) for n = 1..5001</a>
%H A058060 E. Heppner, <a href="https://eudml.org/doc/151410">Über die Iteration von Teilerfunktionen</a>, Journal für die reine und angewandte Mathematik, Vol. 265 (1974), pp. 176-182.
%H A058060 G. J. Rieger, <a href="https://doi.org/10.1007/BF01303534">Über einige arithmetische Summen</a>, Manuscripta Mathematica, Vol. 7 (1972), pp. 23-34.
%F A058060 a(n) = A001221(A000005(n)).
%F A058060 Sum_{k=1..n} a(k) = c * n + O(sqrt(n) * log(n)^5), where c is a constant (Rieger, 1972; Heppner, 1974). - _Amiram Eldar_, Jan 15 2024
%e A058060 n = 120 = 8*3*5, d(n) = 16 = 2^4, so a(120)=1.
%t A058060 Table[PrimeNu[DivisorSigma[0, n]], {n, 1, 100}] (* _G. C. Greubel_, May 05 2017 *)
%o A058060 (PARI) a(n)=omega(numdiv(n)) \\ _Charles R Greathouse IV_, May 05 2017
%Y A058060 Cf. A001221, A000005, A058061.
%K A058060 nonn,easy
%O A058060 1,12
%A A058060 _Labos Elemer_, Nov 23 2000
%E A058060 Offset corrected by _Sean A. Irvine_, Jul 22 2022