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