%I #19 Jun 13 2022 07:34:45

%S 0,1,2,4,9,19,38,77,154,308,616,1233,2466,4932,9864,19728,39456,78913,

%T 157826,315652,631305,1262611,2525222,5050445,10100890,20201781,

%U 40403562,80807124,161614248,323228496,646456993,1292913986,2585827972

%N a(n) = floor(log_10(2^(2^n))).

%C Starting with 2, n successive squarings yields an (a(n)+1)-digit number.

%C Dubickas proves that infinitely many terms of this sequence are divisible by 2 or 3 (and hence infinitely many composites). - _Charles R Greathouse IV_, Feb 04 2016

%H Artūras Dubickas, <a href="http://dx.doi.org/10.1007/s00605-008-0042-6">Prime and composite integers close to powers of a number</a>, Monatsh. Math. 158:3 (2009), pp. 271-284.

%e a(16) = 19728 because floor(log_10(2^(2^16))) = floor(log_10(2^65536)) = floor(log_10(2.003529930406846*10^19728)) = floor(19728.30179583467) = 19728.

%t Table[Floor[Log[10, 2^(2^n)]], {n, 1, 29}] (* _Vincenzo Librandi_, Dec 30 2015 *)

%o (Magma) [Floor(Log(10,2^(2^n))): n in [1..29]]; // _Vincenzo Librandi_, Dec 30 2015

%o (PARI) a(n) = floor(log(2^(2^n))/log(10)); \\ _Michel Marcus_, Dec 30 2015

%o (PARI) a(n)=logint(2^2^n,10) \\ impractical except for small n, but avoids rounding; _Charles R Greathouse IV_, Feb 04 2016

%Y Cf. A001146.

%K nonn

%O 1,3

%A _Jon E. Schoenfield_, May 18 2007