A211661 Number of iterations log_3(log_3(log_3(...(n)...))) such that the result is < 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, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3

Offset

1,3

Comments

For n<16 same as A211663.

Links

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

Formula

With the exponentiation definition E_{i=1..n} c(i) := c(1)^(c(2)^(c(3)^(...(c(n-1)^(c(n))))...))); E_{i=1..0} := 1; example: E_{i=1..4} 3 = 3^(3^(3^3)) = 3^(3^27), we get:

a(E_{i=1..n} 3) = a(E_{i=1..n-1} 3)+1, for n>=1.

G.f.: g(x)= 1/(1-x)*sum_{k=0..infinity} x^(E_{i=1..k} 3). The explicit first terms of the g.f. are

g(x)=(x+x^3+x^27+x^7625597484987+…)/(1-x).

Example

a(n)=1, 2, 3, 4, 5 for n=1, 3, 3^3, 3^3^3, 3^3^3^3 =1, 3, 27, 7625597484987, 3^7625597484987

Mathematica

Table[Length[NestWhileList[Log[3, #]&, n, #>=1&]], {n, 90}]-1 (* Harvey P. Dale, Mar 08 2020 *)

Crossrefs

Cf. A001069, A010096, A211664, A211666, A211668, A211669.

Sequence in context: A204551 A292563 A362367 * A111972 A073458 A194698

Adjacent sequences: A211658 A211659 A211660 * A211662 A211663 A211664

Keyword

base,nonn

Author

Hieronymus Fischer, Apr 30 2012

Status

approved