A211667 - OEIS

%I #32 Jan 18 2024 08:56:29

%S 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,

%T 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,

%U 3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3

%N Number of iterations sqrt(sqrt(sqrt(...(n)...))) such that the result is < 2.

%C Different from A001069, but equal for n < 256.

%H Amiram Eldar, <a href="/A211667/b211667.txt">Table of n, a(n) for n = 1..10000</a>

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

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

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

%F a(n) = 1 + floor(log_2(log_2(n))) for n>=2. - _Kevin Ryde_, Jan 18 2024

%e 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.

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

%o (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

%o (PARI) a(n) = if(n<=1,0, logint(logint(n,2),2) + 1); \\ _Kevin Ryde_, Jan 18 2024

%o (Python) A211667=lambda n: n and (n.bit_length()-1).bit_length() # _Nathan L. Skirrow_, May 16 2023

%Y Cf. A001069, A001146, A010096, A211662, A211668, A211670.

%Y Cf. A087046 (run lengths).

%K nonn,easy

%O 1,4

%A _Hieronymus Fischer_, Apr 30 2012