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.

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, 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, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4

Offset: 1

Author

Leonid Broukhis

Keywords

• nonn
• nice

Comments

From Hieronymus Fischer, Apr 08 2012: (Start)

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

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

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)

Examples

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

Crossrefs

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

Programs

Haskell

a010096 = length . takeWhile (/= 0) . iterate a000523
-- Reinhard Zumkeller, Mar 16 2012

Mathematica

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

PARI

a(n)=if(n<1,0,1+a(log(n)\log(2))) \\ Charles R Greathouse IV, Apr 17 2012

PARI

a(n)=if(n<1,0,1+a(logint(n,2))) \\ Charles R Greathouse IV, Oct 23 2015

Formula

From Hieronymus Fischer, Apr 08 2012: (Start)

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

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:

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

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

The explicit first terms of this g.f. are

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

Extensions

Edited by Hieronymus Fischer, Apr 08 2012

Edited by N. J. A. Sloane, Nov 03 2013