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A010096 log2*(n) (version 1): number of times floor(log_2(x)) is used in floor(log_2(floor(log_2(...(floor(log_2(n)))...)))) = 0. 15

%I

%S 1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,

%T 4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,

%U 4,4,4,4,4,4,4,4,4,4,4

%N log2*(n) (version 1): number of times floor(log_2(x)) is used in floor(log_2(floor(log_2(...(floor(log_2(n)))...)))) = 0.

%C From _Hieronymus Fischer_, Apr 08 2012: (Start)

%C A possibly simpler definition could be: "Number of iterations log_2(log_2(log_2(...(n)...))) such that the result is < 1".

%C Changing "< 1" to "<= 1" produces version 3, A230864.

%C With the only difference in the termination criterion, the definition is essentially the same as version 2, A001069. If we change the definition to "floor(log_2(... = 1" we get A001069. Therefore we get A001069 when subtracting 1 from each term. (End)

%F From _Hieronymus Fischer_, Apr 08 2012: (Start)

%F a(n) = A001069(n) + 1.

%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} 2 = 2^(2^(2^2)) = 2^16, we get:

%F a(E_{i=1..n} 2) = a(E_{i=1..n-1} 2) +1, for n >= 1.

%F G.f.: g(x) = 1/(1-x)*Sum_{k>=0} x^(E_{i=1..k} 2).

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

%F g(x) = (x + x^2 + x^4 + x^16 + x^65536 + ...)/(1-x). (End)

%e Becomes 5 at 65536, 6 at 2^65536, etc.

%t f[n_] := Length@ NestWhileList[ Log[2, #] &, n, # >= 1 &] - 1; Array[f, 105] (* _Robert G. Wilson v_, Apr 19 2012 *)

%o (Haskell)

%o a010096 = length . takeWhile (/= 0) . iterate a000523

%o -- _Reinhard Zumkeller_, Mar 16 2012

%o (PARI) a(n)=if(n<1,0,1+a(log(n)\log(2))) \\ _Charles R Greathouse IV_, Apr 17 2012

%o (PARI) a(n)=if(n<1,0,1+a(logint(n,2))) \\ _Charles R Greathouse IV_, Oct 23 2015

%Y Cf. A063510, A000523, A001069 (version 2), A230864 (version 3).

%K nonn,nice

%O 1,2

%A _Leonid Broukhis_

%E Edited by _Hieronymus Fischer_, Apr 08 2012

%E Edited by _N. J. A. Sloane_, Nov 03 2013

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Last modified May 31 07:14 EDT 2020. Contains 334747 sequences. (Running on oeis4.)