This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A010096 #42 Apr 16 2025 08:42:08
%S A010096 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 A010096 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 A010096 4,4,4,4,4,4,4,4,4,4,4
%N 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.
%C A010096 From _Hieronymus Fischer_, Apr 08 2012: (Start)
%C A010096 A possibly simpler definition could be: "Number of iterations log_2(log_2(log_2(...(n)...))) such that the result is < 1".
%C A010096 Changing "< 1" to "<= 1" produces version 3, A230864.
%C A010096 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 A010096 From _Hieronymus Fischer_, Apr 08 2012: (Start)
%F A010096 a(n) = A001069(n) + 1.
%F A010096 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 A010096 a(E_{i=1..n} 2) = a(E_{i=1..n-1} 2) +1, for n >= 1.
%F A010096 G.f.: g(x) = 1/(1-x)*Sum_{k>=0} x^(E_{i=1..k} 2).
%F A010096 The explicit first terms of this g.f. are
%F A010096 g(x) = (x + x^2 + x^4 + x^16 + x^65536 + ...)/(1-x). (End)
%e A010096 Becomes 5 at 65536, 6 at 2^65536, etc.
%t A010096 f[n_] := Length@ NestWhileList[ Log[2, #] &, n, # >= 1 &] - 1; Array[f, 105] (* _Robert G. Wilson v_, Apr 19 2012 *)
%o A010096 (Haskell)
%o A010096 a010096 = length . takeWhile (/= 0) . iterate a000523
%o A010096 -- _Reinhard Zumkeller_, Mar 16 2012
%o A010096 (PARI) a(n)=if(n<1,0,1+a(log(n)\log(2))) \\ _Charles R Greathouse IV_, Apr 17 2012
%o A010096 (PARI) a(n)=if(n<1,0,1+a(logint(n,2))) \\ _Charles R Greathouse IV_, Oct 23 2015
%Y A010096 Cf. A063510, A000523, A001069 (version 2), A230864 (version 3).
%K A010096 nonn,nice
%O A010096 1,2
%A A010096 _Leonid Broukhis_
%E A010096 Edited by _Hieronymus Fischer_, Apr 08 2012
%E A010096 Edited by _N. J. A. Sloane_, Nov 03 2013