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

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, 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, 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

Offset

1,2

Comments

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

From N. J. A. Sloane, Jun 02 2014: (Start)

The fourth iterate begins as follows:

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)

References

S. Ramanujan, Collected Papers, Ed. G. H. Hardy et al., Cambridge 1927; Chelsea, NY, 1962, p. 128. - N. J. A. Sloane, Jun 02 2014

Links

Enrique Pérez Herrero, Table of n, a(n) for n = 1..2000

R. Bellman and H. N. Shapiro, On a problem in additive number theory, Annals Math., 49 (1948), 333-340.

Example

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

Mathematica

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

Prog

(PARI) a(n)=numdiv(numdiv(numdiv(n))) \\ Charles R Greathouse IV, Nov 16 2022
(Python)
from sympy import divisor_count
def A036450(n): return divisor_count(divisor_count(divisor_count(n))) # Chai Wah Wu, Nov 17 2022

Crossrefs

Cf. A000005, A010553, A036452, A036453.

Sequence in context: A103379 A032550 A339172 * A227533 A235124 A235125

Adjacent sequences: A036447 A036448 A036449 * A036451 A036452 A036453

Keyword

nonn

Author

Labos Elemer

Status

approved