A010096 - OEIS (original) (raw)

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

COMMENTS

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)

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} 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)

EXAMPLE

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

MATHEMATICA

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

PROG

(Haskell)

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