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A036450 a(n) = d(d(d(n))), the 3rd iterate of the number-of-divisors function with an initial value of n.

%I A036450 #34 Nov 17 2022 14:11:10
%S A036450 1,2,2,2,2,2,2,2,2,2,2,3,2,2,2,2,2,3,2,3,2,2,2,3,2,2,2,3,2,3,2,3,2,2,
%T A036450 2,2,2,2,2,3,2,3,2,3,3,2,2,3,2,3,2,3,2,3,2,3,2,2,2,4,2,2,3,2,2,3,2,3,
%U A036450 2,3,2,4,2,2,3,3,2,3,2,3,2,2,2,4,2,2,2,3,2,4,2,3,2,2,2,4,2,3,3,2,2,3,2,3,3
%N A036450 a(n) = d(d(d(n))), the 3rd iterate of the number-of-divisors function with an initial value of n.
%C A036450 The iterated d function rapidly converges to the fixed point 2.
%C A036450 From _N. J. A. Sloane_, Jun 02 2014: (Start)
%C A036450 The fourth iterate begins as follows:
%C A036450 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, ... . (End)
%D A036450 S. Ramanujan, Collected Papers, Ed. G. H. Hardy et al., Cambridge 1927; Chelsea, NY, 1962, p. 128. - _N. J. A. Sloane_, Jun 02 2014
%H A036450 Enrique Pérez Herrero, <a href="/A036450/b036450.txt">Table of n, a(n) for n = 1..2000</a>
%H A036450 R. Bellman and H. N. Shapiro, <a href="http://www.jstor.org/stable/1969281">On a problem in additive number theory</a>, Annals Math., 49 (1948), 333-340.
%e A036450 n = 5040, d(5040) = 60, d(d(5040)) = d(60) = 12 and a(5040) = d(12) = 6.
%t A036450 f[n_]:=Length[Divisors[n]];Table[Nest[f,n,3],{n,6!}] (* _Vladimir Joseph Stephan Orlovsky_, Mar 10 2010 *)
%o A036450 (PARI) a(n)=numdiv(numdiv(numdiv(n))) \\ _Charles R Greathouse IV_, Nov 16 2022
%o A036450 (Python)
%o A036450 from sympy import divisor_count
%o A036450 def A036450(n): return divisor_count(divisor_count(divisor_count(n))) # _Chai Wah Wu_, Nov 17 2022
%Y A036450 Cf. A000005, A010553, A036452, A036453.
%K A036450 nonn
%O A036450 1,2
%A A036450 _Labos Elemer_