A211667 - OEIS (original) (raw)

0, 1, 1, 2, 2, 2, 2, 2, 2, 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, 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, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3

COMMENTS

Different from A001069, but equal for n < 256.

FORMULA

a(2^(2^n)) = a(2^(2^(n-1))) + 1, for n >= 1.

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

= (x^2 + x^4 + x^16 + x^256 + x^65536 + ...)/(1 - x).

a(n) = 1 + floor(log_2(log_2(n))) for n>=2. - Kevin Ryde, Jan 18 2024

EXAMPLE

a(n) = 1, 2, 3, 4, 5, ... for n = 2^1, 2^2, 2^4, 2^8, 2^16, ..., i.e., n = 2, 4, 16, 256, 65536, ... = A001146.

MATHEMATICA

a[n_] := Length[NestWhileList[Sqrt, n, # >= 2 &]] - 1; Array[a, 100] (* Amiram Eldar, Dec 08 2018 *)

PROG

(PARI) apply( A211667(n, c=0)={while(n>=2, n=sqrtint(n); c++); c}, [1..50]) \\ This defines the function A211667. The apply(...) provides a check and illustration. - M. F. Hasler, Dec 07 2018

(PARI) a(n) = if(n<=1, 0, logint(logint(n, 2), 2) + 1); \\ Kevin Ryde, Jan 18 2024