A211669 Number of iterations f(f(f(...(n)...))) such that the result is < 2, where f(x) = cube root of x.

0, 1, 1, 1, 1, 1, 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, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2

Offset

1,8

Comments

For the general case of "Number of iterations f(f(f(...(n)...))) such that the result is < q, where f(x) = x^(1/p)", with p > 1, q > 1, the resulting g.f. is g(x) = 1/(1 - x)*Sum_{k>=0} x^(q^(p^k))

= (x^q + x^(q^p) + x^(q^(p^2)) + x^(q^(p^3)) + ...)/(1 - x).

The first term that equals 3 is a(512). - Harvey P. Dale, Jan 02 2015

Links

Table of n, a(n) for n=1..86.

Formula

a(2^(3^n)) = a(2^(3^(n-1))) + 1, for n >= 1.

G.f.: 1/(1-x)*Sum_{k>=0} x^(2^(3^k))

= (x^2 + x^8 + x^512 + x^134217728 + ...)/(1 - x).

Example

a(n) = 1, 2, 3, 4, 5, ... for n = 2^1, 2^3, 2^9, 2^27, 2^81, ..., i.e., n = 2, 8, 512, 134217728, 2417851639229258349412352, ... = A023365.

Mathematica

Table[Length[NestWhileList[Surd[#, 3]&, n, #>=2&]], {n, 90}]-1 (* Harvey P. Dale, Jan 02 2015 *)

Prog

(PARI) a(n, c=0)={while(n>=2, n=sqrtnint(n, 3); c++); c} \\ M. F. Hasler, Dec 07 2018

Crossrefs

Cf. A001069, A010096, A023365, A211662, A211664, A211666, A211668, A211670.

Sequence in context: A178487 A280560 A044932 * A065687 A300403 A077433

Adjacent sequences: A211666 A211667 A211668 * A211670 A211671 A211672

Keyword

nonn

Author

Hieronymus Fischer, Apr 30 2012

Extensions

Edited by M. F. Hasler, Dec 07 2018

Status

approved