A036450 - OEIS

%I #34 Nov 17 2022 14:11:10

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

%C The iterated d function rapidly converges to the fixed point 2.

%C From _N. J. A. Sloane_, Jun 02 2014: (Start)

%C The fourth iterate begins as follows:

%C 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 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 Enrique Pérez Herrero, <a href="/A036450/b036450.txt">Table of n, a(n) for n = 1..2000</a>

%H 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 n = 5040, d(5040) = 60, d(d(5040)) = d(60) = 12 and a(5040) = d(12) = 6.

%t f[n_]:=Length[Divisors[n]];Table[Nest[f,n,3],{n,6!}] (* _Vladimir Joseph Stephan Orlovsky_, Mar 10 2010 *)

%o (PARI) a(n)=numdiv(numdiv(numdiv(n))) \\ _Charles R Greathouse IV_, Nov 16 2022

%o (Python)

%o from sympy import divisor_count

%o def A036450(n): return divisor_count(divisor_count(divisor_count(n))) # _Chai Wah Wu_, Nov 17 2022

%Y Cf. A000005, A010553, A036452, A036453.

%K nonn

%O 1,2

%A _Labos Elemer_