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 A211664 Number of iterations (...f_4(f_3(f_2(n))))...) such that the result is < 1, where f_j(x):=log_j(x). 7

%I

%S 1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,

%T 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,

%U 3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3

%N Number of iterations (...f_4(f_3(f_2(n))))...) such that the result is < 1, where f_j(x):=log_j(x).

%F 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:

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

%F G.f.: g(x)= 1/(1-x)*sum_{k=0..infinity} x^(E_{i=1..k} (i+1)).

%F The explicit first terms of the g.f. are

%F g(x)=(x+x^2+x^(2^3)+x^(2^3^4)+(x^2^3^4^5)+...)/(1-x) =(x+x^2+x^8+x^2417851639229258349412352+...)/(1-x).

%e a(n)=1, 2, 3, 4, 5 for n=1, 2, 2^3, 2^3^4, 2^3^4^5 =1, 2, 8, 2417851639229258349412352, 2^3^1024.

%Y Cf. A001069, A010096, A084558, A211661, A211666, A211668, A211670.

%K base,nonn

%O 1,2

%A _Hieronymus Fischer_, Apr 30 2012

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Last modified December 7 18:12 EST 2019. Contains 329847 sequences. (Running on oeis4.)